Some training methods that could help:
Praise, reward, and play with him when he chews on the right toys. If he gets bored of a toy and starts to nip at you, try to redirect him to a different toy. Praise him if it works. But if he bites you, playtime is over. Say "no" firmly, then get up and walk away. If he's nipping because he wants play or attention, he will continue to do that as long as it works for him. You need to teach him that nipping & biting only leads to being ignored. Show him the right way to ask for attention by putting him in a sit and playing with him with a toy when he's calm.
For the body sensitivity, I would definitely make sure that he isn't snapping because he's in pain. When he snaps at you while you're petting him, I would also suggest a firm "No" and walking away from him completely. He should be able to tolerate you handling him all over, and he'll learn that if he doesn't, then you won't pet him at all.
You can also try gently touching & handling him in different areas while he's eating or while he's occupied with a toy he really likes, so he can slowly build a positive association with the feeling of you handling him. It can also be useful to have two people: one person handfeeding him food or treats, and the other touching him, so that if he snaps or tries to get away from the touch, the food is immediately taken away. That way he has to choose between getting the treats and being left alone, and if your treats are good enough, he'll make the right choice. Over time and practice, he'll build a positive association with the handling and get used to it.
This is all basic training I've used to teach puppies not to bite & to allow body handling...but if it's a more serious or persistent issue, or if you've tried all this and it hasn't worked, definitely reach out to a trainer with more credentials and get to the bottom of the issue before your pup gets much older!