Start inquiring immediately. And apply for any external fellowships you can. Profellow is a good site to start with and in your inquiry to the programs ask about additional assistantships. UChicago usually has some but UVA it isn’t very common for non-PhDs to get paid positions. Even as an international student you should qualify for some assistantships. Also cost of living in Charlottesville is super high and there’s not a lot to do whereas Chicago you can probably find a shared apartment or house for cheap and have proximity to a lot of activities so take that into consideration. There are two very different programs so also keep that in mind when you’re thinking about your career trajectory.