What can I do if I’m experiencing homelessness?

I am currently in a motel with my dog. It is just me and her (she is a husky).I have been calling around to shelters but a lot do not have any vacancies. What are my other options? I have never bought an apartment or any time of housing so I am not sure if I qualify. I currently only have a part time just I just started this month and have been doing side gigs to make money (dog sitting, dog walking, baking, babysitting, cleaning houses, etc). I have about $5000 saved up. It’s all I got and my income isn’t stable. It’s a lot harder now that I do not have a home. I am in desperate need of somewhere to live. If you guys know of anything I can do that can help me and my girl ASAP I would greatly appreciate it. Thank you.