Some of mine went full time and my last went part time. Mine started earlier just because of our situation, so at 3 they are a bit more aware of what's going.
Perhaps a bit harder where your little guy had to say bye to a babysitter - changes.
I found with one of mine, if I stayed and chatted, and there was one toy he looked forward to each day and the teacher let him bring one home (special) and he brought it back ... maybe such an arrangement could be made with the teacher. Or something you could put in his pocket to remind him of Grammy, that he could take with him. One of mine had a little toy he kept in his pocket, another a little message from me.
It was just comforting. Similarly, if I showed up and chatted and just stayed (no drop and rush off), he felt I was 'ok' with daycare and really liked it - so he felt much more comfortable with the place and he loosened up. It depends on the child. Sometimes they encourage you to run off - because kids do better. Other times, having you stay for a while, and make it shorter and shorter .. is better. Talk to the teacher - maybe even visit with him on a day you have off where he's not scheduled and just sit with him and listen to story time. Maybe say "now wasn't that fun".
I don't know - they would have suggestions. Sometimes staying is not the thing to do. My mom taught kindergarten and by that age, they didn't want parents there. But at preschool age, it was ok to stay for a while on the fringes. They would nod when we were supposed to scoot out.
If dad is rushed at drop off/pick up though and not talking to the other moms (my husband was our pick up person) sometimes the kiddos feel more anxious - because the other mommies might be standing around chatting for longer. So that's a thought. My husband used to run in grab the bag and go. He is super friendly, but he had worked a long day, and didn't get it didn't hurt to chat for a while. So the teachers would chat with him to encourage him to stay longer. It helps the kiddos.
It does get better though. As for the wetting himself, maybe the sitter used to remind him and the daycare teachers don't have time to or they just have a different system. Some kids only just learn at 3 (like mine) so don't think this is totally abnormal. To regress during a change - pretty common. It will come :) So long as they have a change of clothes for him, I'm sure they are used to dealing with it.
If you're anxious about it of course they pick up on it - so try to keep it fun, maybe have a special stuffy in car for him to look forward to when he's picked up (Bunny is waiting in car!) and nothing with special treats/rewards - like a special show to watch when he gets home from daycare to look forward to.
I used to have a picnic snack waiting at the end of day sometimes while I made dinner. We just had 'routines' the kids loved on those busy days. Earlier bedtimes - because daycare is exhausting. It's go go go and it is a lot more tiring than at a sitters - so if he seems more emotional, totally normal.
Good luck and don't worry too much. That too, is very normal :)