Is this normal?

My kids (DS is 6 and DD is 2 as of August) and I recently moved into an apartment by yourselves. We lived with my parents for over a year and a half prior to this. Ever since the move, I feel like my daughter is regressing. She's potty trained during the day and has been peeing without any word at all. I've had to start putting pull ups on her during the day. She has also been crying an asking for her pacifier again. She hasn't had one since before her 2nd birthday. She even tried getting me to go to the store to get one. She's been super emotional lately also. The first night here she cried and cried begging for me to take her back home. I was wondering if this was a normal thing to happen because of such a big change. Should I give in to her paci requests until she has adjusted to being here?

It sounds fairly normal to me. Some kids handle change better than others. I had a similar situation when dd was a baby - when we moved, she stopped sleeping through the night. I think she was just scared because the new environment wasn't familiar to her.
She probably just wants the pacifier for security. She's in a new place and that would make her feel like things are more "right," maybe. I would try not to give in though. Just make sure she knows she's safe and this is her home now, and emphasize all the good stuff about your new place. She'll get used to it.
I moved 800 times as a kid and it never got "easy" but it forced me to learn a lot about resilience and how to adapt.
Are you still close enough to your mom that you can go visit fairly often? That might help her understand that she didn't actually lose anything.