Hi. Are you in the US? I'n guessing so as the food is sold by Petsmart
It's great that you've adopted this little guy and we'd love to help get things right for him, so please don't take any of the following personally - but some changes need to be made for his wellbeing.
I looked up Full Cheeks Hamster food and I think that is probably the main part of the problem. It's really important you get a different mix.
Firstly these are Timothy Hay pellets and totally unsuitable for a hamster unfortunately - while Guinea pigs can cope with this much roughage, hamsters can't. And that probably explains the sticky poo mess (which shouldn't be there - they should have small little black seed type poops you can just pick up easily in your fingers).
So it sounds like he might have diarrhoea or digestive upset as well, which can become quite dangerous.
If it's the food I've linked then one pellet a day isn't enough and it would be underfeeding him which he is probably trying to tell you. It does say in the reviews that they are quite big pellets but most reviewers seem to be giving a few of them.
The ingredients aren't good for a hamster anyway (it sounds similar to Oxbow which is also bad for them).
They really need a muesli mix as that's more normal for them and yes they do pick things out and not eat all of it but they still get the nutrients required and it allows a happy hamster and more normal behaviours. You can even do a tiny bit of scatter feeding at the same time as putting some in his bowl, with a muesli mix and he'll really enjoy foraging for that.
It's really annoying that they sell these things full of Timothy Hay for hamsters, but then it's usually better to get a Hamster specific food rather than one for Hamsters and Gerbils (who have a completely different dietary requirement).
I don't want to alarm you but we had one member's hamster who starved to death - because she hadn't been giving him enough to stop him hoarding it, and then removing his hoards when cleaning out. He had also been frantic before he died. They can also start eating their poops if they're short on food (which is actually normal behaviour as they have two stomachs and can redigest vitamins etc from their poops). They're hard wired to do this in case of food shortages in the wild. But if he has diarrhoea he can't even do that. So sorry if I've misunderstood, I was just worried he might not be getting enough food and he does sound to have loose poo.
I notice reviewers said hamsters really enjoy it as well (but it's still very bad for them).
I think it will make life a whole lot easier for both you and the hamster to use a better mix - which doesn't necessarily cost more.
According to some of our US reviewers, the best hamster mix in the US which the right contents, sugar free and good ingredients, is Higgins Vita Garden. Which is a bit low in protein so needs extra protein treats to supplement it.
If it was me I would probably just use Harry or Hazel Hamster which includes everything needed and no supplementing necessary - which is easier for some owners and I've used it for our hamsters, who have all lived long lives. The pieces are the right size for a Syrian hamster as well and they like the food. It has whole sunflowers and peanuts in as well as some biscuit and pellet type things. It contains all the correct nutrients and vitamins and is sugar free. Most of them leave the brown pellety things in it, which doesn't really matter as that's the fibre element and they get enough fibre from the rest or from a bit of daily veg which they should have too.
They absolutely love a bit of fresh veg daily and that will make your hamster happy too. Just a really tiny bit - about 1cm cubed size (although can be any shape - that;s just to give an indication of size). Cucumber is a favourite as is broccoli. And both of those keep at least a week in the fridge. Also raw carrot. Those have been my staples, rotating them so it's different each day. And carrots keep for weeks in the fridge (although not their favourite veg!) Cucumber and broccoli seem to be the favourites. Ours look forward to it every day and it's the first thing they go for. And they actually eat it rather than hoarding it as well so you don't need to worry about it going rotten in the cage (which is why you only give a very tiny amount). I have never yet known a hamster hoard fresh veg.
I don't know where you'd get the Higgins Vita Garden from, but I know you can get Tiny Friends Harry Hamster or Hazel Hamster from Amazon online. They are both identical mixes, but Harry is the Uk marketed version and Hazel is the US marketed version. The labelling is slightly different due to different legislation for different countries.
But it's an all in one mix but also has lots of variety which they like. I would suggest switching his food asap and hope his bowels settle down, but keep an eye on that as they can get dehydrated really quickly. So best to check his water bottle is working daily (I tap my finger under the spout to make sure water is coming out).
But introduce the veg gradually or that can give them diarrhoea if they're not used to it. So give it every third day for a week and then you can go to every day. But honestly it's their reason for living with our hamsters! As soon as they smell it they're up and eating it.
Best not to give cabbage or lettuce which is too watery (there are varying views on whether that's ok occasionally but I don't risk it and they like cucumber and broccoli anyway).
So hopefully that will see some improvements. You'd need to give about a rounded tablespoonful a day in his bowl and also try scattering a few bits of it on the substrate as well so he can enjoy foraging for it - they really like doing that. But I wouldn't scatter all of it as if they don't want all of it it can make the cage messy so best to put most in the bowl still.
The only thing is I've just noticed Harry and Hazel Hamster seems to be ridiculously priced or not available on Amazon right now, but you can get it here ($11.99 for 2 bags).
2 PACK Supreme Tiny Friends Harry Hamster Food Tasty Mix 700g Sugar-Free Peanut | eBay
If he likes his ball and its supervised then that's fine. If you can at some point, you could maybe try the bathtub taming. Or even just do playpen taming if you can set up a playpen.
You haven't had him long anyway so I think once you get the food routine sorted, he'll be happier. But if the cage is a bit small or there isn't enough substrate, he will bar chew.
One thing I'd suggest as a good investment is a corner potty litter tray. They will use it and then the cage stays clean and dry. They nearly always choose a corner of the cage as a toilet as you've mentioned, so if you put the potty litter tray there, with Chinchilla bathing sand in, then I am absolutely sure he will use it! All ours have. It really makes life easier for both of you. The sand soaks up the pee and the cage stays clean and dry. So you just take the litter tray out and empty it and refill it once or twice a week (not every day - not necessary). And don't have to mess with his cage too much. He will pee in it. Their poops they tend to do anywhere, but they aren't dirty or smelly - (as I mentioned they even hoard some for emergency food supplies or eat the odd one) - they should like little black seeds almost. So don't worry about those, unless there start to be a lot around, then you can just pick them out (spot clean them out).
Spot cleaning is basically just removing a handful of substrate in an area where it's perhaps a bit poopy, and adding another new handful to replace it. Then mix it in a bit so it still smells familiar,
Using the potty does mean buying sand as well as substrate but it lasts for ages. It must be Chinchilla sand not "dust" as that's bad for their airways.
This is the right size corner litter tray (be careful some online are huge!)
Amazon.com
Some people use the tiny Friends chinchilla sand. Others use childrens play sand but that needs baking to sterilise it. Some reptile sands aren't suitable though.