In that case I wouldn't try and do it yourself.
It sounds like he really does need vet treatment as his teeth either sound loose or misaligned. He is three years old and that is very old! I am wondering if his teeth are actually loose and some may fall out. The top teeth are quite small. I've had them fall out before in an older syrian. I maybe wouldn't worry about the top teeth so much as they are quite short. That one might fall out.
The long bottom teeth are more of an issue. If a bottom tooth gets too long it could puncture his palate at the top and cause severe pain and even brain injury. It's when that happens their mouths can't open. But also sometimes if the bottom teeth get too long they can't close their mouths either.
Yes if you cut them wrong you could cut down too far and leave him with severe pain if it cuts the nerve.
If the bottom tooth sometimes sticks out the side or sometimes pokes back it sounds like that one might be loose as well. What is the other bottom tooth like?
Definitely give him soft food so he's getting the nutrients (and the pleasure of eating easily). Plus the hard hamster mix in his bowl as usual. I'm just wondering, if you give him a tasty hard chew stick (ie one with food on) if the loose teeth might just fall out?
I think he might need some pain relief as well at that age and if his teeth are moving around (it could cause pain) and that is another reason to see a vet - so they can prescribe some metacam.
I would try one of the vets in your area again, and say can you do tooth extraction? And prescribe metacam. I don't think they could extract a tooth without an anaesthetic though and that is risky at his age.
Yes do upload a photo!