Did some more testing and I got the board connecting to my AP again. (One with a stronger signal than the original one that I had been connecting to.) This test yields the same result as I was getting before. I connect to the AP then the test runs for a bit and hangs. (It stops producing serial output as well as stops responding to pings.)
I'm trying to set up an AP right next to the board to eliminate any possible interference problems, but I'm having some of my own network challenges.