Thanks for the ideas. I had all of the hoses replaced and the fluid flushed 3 years ago when I bought the car. So I have been hesitant to blame this problem on the hoses. When I remove the pads, the rotor is loose, so I don't think the emergency brake is too tight. All of the major brake components are original if that helps.
I can push the pistons back in their bores by hand using the pads. The pistons go in fairly easy but push back out in about a second. The pads aren't so tight that they lock up the rotor, but they are snug on the rotor. Since the caliper is new, I thought that it still may have air trapped. The way the pistons moved back almost seemed like trapped air. I bled it again two different ways. Using a vacuum pump and then the two man method with the brake pedal. I got some air with the vacuum, but absolutely none with the two man method. So I think all of the air is gone but the pistons still spring back.
I remember that when I exercised the front calipers, the pistons stayed in the bore and didn't push back on their own. That seems normal. It almost seems like there are springs pushing the pistons back out on this rear caliper.
When you disconnect the brake line to the calipers, is it normal for the fluid to continuously drip? The line to the rear caliper did this until I put a rubber plug in it and found a new caliper. It wasn't any pressure, but just a constant drip every second. I don't know if this may be related to my problem.
This is a frustrating problem. Thanks for your help!