Problem is, two of the handsets have stopped reliably
communicating with the base. I often can't get a dialtone
to call out, nor answer an incoming call. Error message
sometimes comes up on the handset display, saying I'm
out of range.

Well, there's your problem.

The most-likely cause is that the output of the base station is down. Make
sure its antenna isn't dirty (!!!) or broken, and position the base station
where the handsets can easily "see" it (RF-wise).