I got it!
I don't know what went wrong before, but I am now able to save AND load with the cassette interface. I connected the Tandy to my modern Dell laptop and saved a file. First I tried a text file. I recorded and saved the data sound as a .WAV file on the dell, and then copied it onto my cell phone. I opened the TEXT word processor on the Tandy again to get a blank file, and loaded the sound from my cell phone. It worked! The text showed up on the screen.
I then tested saving and loading BASIC programs the same way, and it worked without any problems!
For some reason, if I try to play the sound directly from my Dell laptop, the Tandy won't load it. But I had the same problem with other vintage computers using a cassette jack. It must be just something about how the Dell plays sound files, these old computers don't seem to like it. But it works if I play the sound from a cell phone.
I'm just so glad that now I can use the little Tandy computer and save files externally and not have to worry about losing my work if the batteries die!