Both tires are flat, and it seems that when I fill with 20 psi air, they don't seal on the rims. I don't think there is any kind of hole/leak in the tires themselves.

Had the same problem with mine a few years ago. I used a ratcheting motorcycle tie-down strap around the circumference of the tire to expand it against the rim. (Note: You may want to take the tire/wheel assembly in the house where it's warm to do this,. the tire will be more pliable). Then I used an air compressor to fill the tire while at the same time releasing the strap. You really need a compressor to put the air in fast, not a hand pump. Also, I used that green slime stuff in the tires to seal against further leakage. They are tubeless tires and it is common for them to get pin-hole leaks in the tread area... probably made in China. I haven't needed to add air since then.

Thank you. Tried most of that already, but didn't think of bringing it inside to the warmth. At this point am probably going to venture out to a service station and have them do it on their machine. I just can't get it to seal...

I'm not sure what a service station can do for you. When I was a kid and worked at a service station (many, many years ago) we did it with an inflatable band around the tire to make the tire bulge out and seal against the rim... similar to the ratcheting tie-down strap I suggested. The inflatable band was probably capable of exerting even more pressure on the tire. Good luck!