Hi A.,
I know this is very late, but you and your daughter just seem so sad that I wanted to reach out and give you a virtual hug.
I don't know what your work schedule is, but you definitely need to work some time in for your daughter. When you are changing your son, have her bring you the diaper and the wipes. When you pick her up from daycare, go for a walk, your son can be in the stroller, it doesn't have to be completely one on one, just make time to do things with her or with both of them. If it's too dark now afterschool, do this on the weekend. Once we have a dry day, pile up a bunch of leaves and let her play in them, your son can be in the bucket car seat of stroller outside with you, and you and your daughter can throw leaves at each other. Get out and LIVE!!
We didn't have the financial stresses you are under, but my daughter is highly prone to "cabin fever", if we don't get outside every day, it affects her whole well-being. (I am a full time mom who travels a lot for work, and until this year, my husband worked away from home a lot as well, we live in a rented apartment that we have more than outgrown -just giving you our background so you don't think my daughter has a silver spoon in her mouth. We're working hard, just not struggling.)
Make sure you read her a story everynight at bed time. Adjust her bed time or your son's, so that you have some one on one time for her then. During the day, have plenty of art supplies that she can draw on and color.
You don't need money to show your daughter you love her. I'm sure you are sad, stressed out, and depressed too, and your daughter is feeding off this. Be sad at work, be sad at night in your bed, talk to a therapist, or friend, BUT, start living for your daughter. Don't let your son be an excuse why she gets neglected, she's only 3 she needs you at least as much as he does. Don't let your husband's internship be an excuse for him to not step up and do his part. Both of you need to take care of your kids first, everything else WILL fall into place.
Let the house work slide a little, or give your daughter things she can help you with (this will cause the whole job to take a lot longer, but she'll be doing things with mommy), SING, be silly.
If there are kids in the daycare that live near you, have them come over or meet in a park for a playdate with your daughter, if she starts to play wiht them one on one, she'll be more likely to play in daycare.
I don't think you need to change daycares so long as the one she is currently in is clean and safe, has plenty of toys/puzzles/books/etc. to provide a stimulating environment, and so long as it has the right ratio of adults to kids, and the caregivers seem to be engaged. I think as she adjusts, she'll make friends and get used to the teacher. Now, if you think the teacher, sits on her bottom all day and lets the kids do their own thing, instead of directing them toward various activities, then it's worth pulling her out altogether or trying to find a way to pay for the old one, even with the religious issues that go with it.
Again, get her to bed on-time, and get 15 minutes to a half hour outdoors with her everyday. Sing songs, and be a kid with her. Use that time to put the rest of the world aside. It will do you both some good.