So my advice would be: don’t do both at the same time.
Travel is a pain tho. Especially if you have pets. I've got a cat who hates traveling, and we are best friends, so it wouldn't be right to leave him alone constantly. Ah well.
Awww! I'm in a similar situation. Have you considered getting a small camper van and customizing it for the cat (litterbox, water dish, food station etc) and travelling around the country in that. Would save on hotels also and once you are done, you can sell the van and made back most of your $ invested in purchasing it. I'm toying with this idea.
Have been in NZ for 6 months but I'm still young so I have lots of working holiday visa options available still, which makes it easier to get bank accounts and stuff in a new country for freelancing. I feel like the logistics of moving around often would get annoying quite quickly
Work, then enjoy the benefits of disconnecting from it as you travel.