After some troubleshooting and help from Jason/NT7S and Robert/AK6L, the problem turned out to be the transformers. Apparently they were step up/step down transformers and not the 1:1 transformers I really needed. Ordered some new transformers from Mouser and put them on when they arrived.
Still ran into problems getting the radio to transmit a signal, but after a closer examination of the board, I discovered I had connected one of my ground wires to the wrong spot so the audio signal to the radio ended up going to ground instead of the radio. Duh.
Fixed that, connected everything up and was able to make my first digimode QSO using the new interface with KC8MGD up in Michigan. From where he was, I had a good signal and wasn’t splattering across the band, so that’s a good sign.
So excited that I got this working! Now I can work on either coming up with an enclosure for this prototype board or build v0.3 and try to make it more compact. I have two more transformers that I can use to build a second interface, which I might save for later once I’m finished messing with this prototype board. I think I’ll try to find some PS2/mini-DIN6 sockets so that I can plug/unplug cables to it and get a little more flexibility about what I can plug into the board.