I think the first step would be to ask for help. Have you talked with your music director or production manager? There's only so much one person can do. Is there an assistant director, intern, extra chorus member, understudy, or someone else who can take on this task when actors are absent? Your PM might be help you figure out who's available.
If you talk to your MD or director, see if you can pick out times when it's most important to have you "reading in" during songs. For instance, during choreography or blocking sessions, someone else could read in the missing part, but when running a number all the way through, you would read it?
For blocking/choreo when no one is missing, can you play a recording? Also, most of the music directors I've worked with can feed in the lines well enough when choreography is being taught- even if it's not every word, they can get enough in for the actors to follow along. Have you asked them if they are able to do that for choreography sections?