Well, guys, I got the Uniroyal tire today (fast shipping!) and it was exactly what I needed. The Columbia had two of these exact Uniroyal tires on it when I bought it. Both appeared to be in like new condition - zero tread wear, no cracks, mold flash still attached, etc. BUT the front tire was damaged, as though the bead was pulling loose. It wouldn't sit straight on the rim.
After mounting the new tire, the old three speed is finally finished! Took a spin up the road, and she's stable as can be - truly an amazing ride for a 54 year old bike!
Thanks again for all your help and advice, guys.