Hi
We were in a similar position to you but, instead of the two dogs, we had three kids under 8. In the end we had to go with another sh**ty rental at the top of our budget for 6 months and that gave us a little breathing space. It's no fun moving truck loads of furniture every few months and breaking leases left, right and centre but we have now found somewhere fab where hopefully we will stay for the foreseeable future.
Hope you find somewhere soon