So with my LO, I found that keeping things calm for quite a bit before the nap was also necessary. Before going into her room and swaddling her, I would walk around the house closing curtains, singing the same songs softly for around 10 minutes. Then into her room to swaddle, sing sleepy song, sound machine on, lights off, out the door.
For nights, we do bath, massage (really helps to relax them), nurse, then burp, swaddle, sing sleepy song, PD , lights off, leave room.
The first step towards independent sleep would be weaning nursing to sleep. If you don't mind doing it, its fine but if it is affecting her ability to resettle at NWs etc, then maybe it is time? Then we introduced the 4 S wind down described by Tracy. In the beginning, I would hold DD till her eyes started closing and then PD. Gradually I PD awake. So it is a process. Hopefully she will find tummy sleeping appealing. After weaning the swaddle, I found DD sleeping tons better that way.
Also regarding using the swaddle for naps -- when we weaned, we had the same issue at nap time with her hands disturbing her. We had to make her room absolutely dark and we put a heavy folded sheet on her tucked in on both sides of the mattress. This way her hands were away while falling asleep but she could bring them out later.