Thanks for the advice, guys. It's not too far off from anything that I was thinking, on all counts.
I got home from work this afternoon, and I slowly knocked both races out of the stem tube without any issues. There was a washer under each race (between the frame and race), as I was expecting. The races were definitely left in the frame when it was powder-coated, as the washer under them and the edge of the race both had some powder-coating finish on them. I was surprised how easily they came out despite this. The bottom bearing doesn't look too bad, but the race on the upper is shot. I'll just replace both since I've already got it apart.
The bearings I pulled out are SKF model "32005X" which have dimensions of 25mm i.d. x 47mm o.d. x 15mm depth. They seem to run about $16-$24 (USD) for SKF brand and cross-referenced brands. This size is used in several automobile applications, as Advanced Auto and Auto Zone informed me that they either have them in stock or can order them. I haven't bought any, yet, as I won't need them until the middle of next week or later (I have my front rim out being straightened
).