Hello,
I have lived in RI for quite some time on and off. We had a lot less trouble finding an apartment that would accept pets this last time we moved here. Before you give up your dogs, take a good look around!
You may have to pay a pet fee, but it's better than giving them up.
My family lives in a complex in East Providence and it actually shocks me the size dogs that people have here. We have people living here with a French Mastiff, two cats, and two birds!
I also have a friend who found a private landlord in Providence who allowed her to have a dog in the apartment.
So, you may have to look around a little... and you may have to pay a little more... but you can definitely find a place that takes pets! Don't give up hope!

Check out the adds in projo.com to start.