How could I put out a request for proposals for a longer-term foster type stay (li 1 to 3 months)?

I am in the process of being divorced and we will likely be selling our house. We have two dogs that will likely stay with me. When the time comes to sell the house, rather than me having to scramble to either find a house lease (and likely be locked in for 12 months) or scramble to buy another house, I would like to either find a place for us to stay (of course paying fairly) for up to a few months, or find a place where I could pay to basically foster them while I stay in a short-term apt until I can buy another house that is appropriate for us. This could happen in around 2-4 months when I will need this.

Thanks Joe