It's true that you may not get a house anytime soon, but don't worry, you're in good company with everyone in our generation and more than half of the millennials in this country. I'm not sure of the specific program you mentioned, but it seems like you're doing everything you can and the hard work will eventually pay off. The market will go down at some point and you might also be set up easier with papers by then. RI just approved driver's licenses for undocumented folks so maybe you could start there, i think it kicks in Jan 1 2023.

You can always try to get a second job, or look for a better paying one, but don't let work get in the way of living your life! You have no debt and it seems like you are fine getting food on the table and having a bit of disposable income on the side. It's probably not what you want to hear but why not enjoy things as they are for a few more years while steadily saving for a down payment? Ask yourself what is the most important part of owning a house? Is it building wealth, control over your living space, etc and see how you can approach whatever is most important to you while you are still renting. Just my 0.02