If you're on a regular backpacker's budget but decided to bring your fur baby after seeing how possible it really is, finding places to stay free will allow you to have the same length of stay with your budget for one. Here are a few options to consider that will provide free stay (and sometimes food):
Trustedhousesitters.com is a place to connect with people from around the world looking for house care or petcare while they're on vacation. While caring for their property, you will typically have access to laundry, a full kitchen, internet, and at times even a car. There is a monthly fee of €8.25 but when you weigh the benefits, it's a worthy investment.
At Couchsurfing.com you'll likely find the largest assortment of pet-friendly accommodation. There really is no catch to couchsurfing. It's simply a community of travelers or people who welcome in travelers, often showing them around and including them in their local lifestyle. Do be careful with couchsurfing though. Don't enter in a situation that makes you feel unsure or uncomfortable and always read people's reviews of potential hosts.
My favorite free stay source is Workaway.com. With a workaway, the catch is that you are required to do work in exchange for your free stay (and sometimes food) but the work gives you a sense of purpose and you're often joining a community. There is an annual fee of $29 for a single person and $38 for couples. Workaways are an excellent way to meet people and have time to really become a part of a place and feel like a local.