What should I do to find an apartment?

I’m trying to find a place for my boyfriend and myself ASAP. He is about to be homeless and things are getting worse at my house. Unfortunately, I have a small dog and a lot of apartments aren’t pet friendly. In addition, I only have a budget of about 600 a month (since we are splitting it in half, that would be around 300 each) I tried to apply for housing assistance, but they have a wait list of two years, which really isn’t helpful here. I’ve found a few good apartments (we’re searching in Albany, New York) but he is extremely picky. We’re looking for a pet friendly, one bedroom apartment in the Albany area with a budget of around 600 a month (700 maximum). I’ve been looking for places online, but so far I have only found places that are too expensive, no pets allowed, or that don’t reply to me. I’ve tried Craigslist as well as realtor sites.
Any ideas?