Hi Melissa,
I've been in your shoes and COMPLETELY understand! I also second what the last two posters said.
The state doesn't need to know about this center unless you tell them. Everytime you submit an application online is proof that you looking for employment. Keep all of the "receipts" of applying online. They will only know if you really trying if they run an audit. That is why it is always smart to keep a record of everything.
I would not include this daycare or any other in your record keeping if it doesn't feel right. I WOULD call the daycare that is offering you an interview/position and tell them that you do not want to waste their time and that you may have another lead that would better fit you. That way you do not burn a bridge you may need later, and you never know who is related to whom and what may be said about you if you completely disrespect them. You need to be professional.
You should really look at being a nanny. The pay is usually better than a daycare and you only watch a couple of kids in their home. Try www.care.com to advertise yourself for free. Also look at local papers and put up creative and professional flyers that offer your experience.
If you find yourself close to being out of unemployment benefits and haven't found the right fit, consider calling this daycare to see if they may still be interested in you.
GOOD LUCK! I know how hard it is to do right by your child and make it all work out.