Hello. You have done so well with this little hammy and it is great that you took her to the vet. You have done a lot for her.
I know this has been a bad experience - but she doesn't hate you - she's scared and defending herself because she doesn't know what you are trying to do and probably just wants to be left alone for a while to settle in (especially if you haven't had her long). So now is the time to pat yourself on the back, find the best compromise for you both, and accept there is only so much you can do - the rest she can decide for herself.
Is it medicine in the syringe or just food? Whichever - the best thing you could try now is to leave it out very near her nest in a little dish (jar lids are quite good because they are low and easy to get at). Put some in the lid and she will take some from it as an when she needs. Getting anything into a hammy 3 times a day is difficult! They eat when they need to but she probably will eat at least twice a day. You know what they say - you can take a horse to water but you can't make it drink. Hammies can be a bit independent and will do the opposite of what you want! But if you leave it very near her nest area I am sure she will take some now and then.
Souffle is right the vet shouldn't have asked you to syringe feed her, especially as you have only just got her so she is not used to you yet - but our hammy wouldn't accept that either and he is tame and used to me.
She will settle down and stop feeling stressed. Leave her alone for a couple of days now. Make sure she has lots of torn up strips of toilet paper for nesting and somewhere dark to go in her cage (eg under a shelf or a large-ish house). Make sure her food and water are very close to her nest (if she can't walk then right next to her nest within reaching distance).
She will forget and forgive so she doesn't hate you and it wasn't personal
Your feelings and confidence have taken a knock and yes it is stressful! Do something nice for you to feel better and just talk to her through the cage in a gentle tone to reassure her - it will help you both feel calmer and better. I always say sorry too lol. And then explain why I had to do it. I'm sure they understand!
Try and understand it from her point of view - she has had a massive change - moved from one place to your home and probably a new cage as well, which she may not have yet started feeling familiar with. Then she's been to the vets (a stressful thing for hammies) and then this syringe keeps attacking her! So she's attacking it (I shouldn't even try and use a syringe again - she sees it as the enemy). With a new or untame hammy they often "see off" anything that enters their cage. Sometimes they are better to do things with out of the cage when they are less territorial, but if she is sick then taking her out of the cage isn't a good thing to do right now probably.
You don't mention what kind of illness - is it diarrhoea? Sometimes you need to find a balance between treating them, letting them heal in their own way, or letting them go in their own way. The positive I can see is, the vet thought it was something that was treatable as they didn't suggest putting her to sleep.
Your little hammy is not ungrateful - she is very lucky to have you - she just doesn't know that yet.
Make her comfortable - lots of substrate, loads of torn up strips of plain white toilet paper (put a heap very near her nest/house so she can reach them without coming out if necessary. If she isn't well enough to build a nest, then put some gloves on and arrange the paper around her like a bit of a nest. Something like a shoe box can make a nice sized house/nesting box. Cut the bottom out and a hole in the front and pop that over her and her nest. She'll feel more secure in a house she has a bit of space in that's dark inside. Then food and water right near the house. Then all you can do is see if she comes out now and then (maybe she will after the lights are out at night. It's amazing how a tasty treat can help them forget everything and their stresses. So if she is mobile at all put the odd tasty treat near her house to tempt her - a small piece of cheese or cucumber.
Don't let this put you off. And try not to worry about everything at the same time right now (easier said than done). It feels like a disaster with knock-on effects - but time heals everything one way or another. So don't worry about what you may or may not be allowed to do in the future right now and don't worry about what your husband thinks about hamsters - maybe you can have two - if you want to show him how they are when younger and active. Don't worry, just do your best and leave your hammy to work things out her way.
Apart from the above mentioned and putting the food/medicine out in a dish once a day and change the water, I would leave her alone completely for at least 3 days now. Keep an eye out to see if she comes out. Don't clean anything for at least a week and then, if it's not too bad, leave it another week (or just take a handful of pee'd on substrate out and replace it with another handful. Don't touch her nest or house. I guess she may pee in her house if she's not very mobile, but still leave it a week, and then come back on here and we can see what's what. In fact come back on here every day anyway if you want!
You have done a wonderful thing adopting a 3 year old hammy and giving her a home and trying to get treatment for her. It's probably best to put softer foods out as they can have problems eating hard food when they're older. So soften the hamster mix with a bit of boiled water in the jar lid, wait till it's gone cold and then put it out. You could also put some baby food out in another lid near the house too. They need the hamster mix most for the nutrition, but the baby food is nice for them.
I get 4 months age baby food which doesn't contain things that aggravate a hamster's stomach, but still check the labels - so the need to have no onion or garlic or leek, or spices, no tomato and no lemon juice. So it tends to limit it to savoury ones and I find most 4 months ones don't contain any of those. Some do so just check the labels.
It is a horrible thing to feel attacked and bitten but it's not the hammy, it's her fear. She will get more familiar with you from the sound of your voice when you talk to her through the cage. And you could also do the tissue trick. Put some sheets of toilet paper up your sleeve or down your bra and leave them there for a couple of hours. Then tear each sheet into strips for nesting material and put them next to the house (or if she can't move to forage for them, pop them in the house next to her with gloves on). That way she will get used to your scent and feel less scared.
Let us know how it goes. The important thing really right now, is that she is happy for as long as she has, rather than completely cured xxx Sometimes they are happiest just being left alone and you can talk to her. If she seems to be in a very bad way and really suffering though - well let us know how she is doing.