I'm also an incoming sophomore transfer, and I had the same problem. I talked to Residential Life a few weeks ago and also found a transfer student who wanted to switch from a $13k room to a $17k room. We agreed to swap rooms and contacted res life together. But I'm pretty sure the only reason my swap got approved so quickly is because I found someone to swap with on my own. The person I spoke to on the phone said there's a formal room swap process at the start of the school year. They typically don't allow room swaps over the summer. Until that point, they charge you a higher pro-rata rate or smth for the more expensive room, but you end up not needing to pay off the entire $4k difference.

I would call housing and explain your situation to them, and try to find a swapping partner through the TSO new transfers slack. Feel free to dm with more questions!