$350 a month with pets isn't happening in a nice area. Maybe you could get a ROOM somewhere for that.
Cheap in a nice area do not go together. Only thing is perhaps a room but the dog will hinder you.
Contact Section 8 or your local Housing Authority. Keep in mind that waiting lists can be long depending on where you live. But, it never hurts to ask.
Need a place that allows small dogs too. Need as cheap as possible, in a nice area, not too rough if possible. Preferably in Chillicothe but please list all in Ross County. Also if there are any realtors or local sites to look for apartments, houses etc for rent. and rentals that are privately owned too. Not the main ones like apartments.com etc. They always list the same ones.
