Scan one copy of each for a digital archive, then toss the sheet music.

Just bear in mind as you go that what is dreck to you may be judged usable by 1) future church musicians or 2) future academics. Look at the doctoral theses that have been published about the most trifling music in surprising places. “Cathedral Music in a Rural German-American Diocese at the turn of the 20th century” may be an excellent dissertation for someone 30[0] years hence to write.

I would do the following:

1) Scan to PDFs at 600 dpi, including any marginalia. Put a digital copy (CD and/or thumb drive) with the paper copies, in addition to any online storage, which may be lost if the diocese switches IT vendors, etc. Consider uploading to CPDL/IMSLP later too, to benefit others outside the diocese.
2) Keep 3 copies of each work, in a suitable environment in appropriate, organized acid-free folders. That way there is an archival copy to keep in storage all the time, a working copy to be given to anyone who would like to study the piece, and a spare in case the working copy is destroyed.
3) While you’re at it, make an Excel spreadsheet listing title, composer, date, source within the diocese (if known), and number of copies present when collection was culled. This would let future researchers discover that X Mass only had 8 copies, and probably was not sung for a big event by a big choir, but Y Mass had 100, and thus Y would be the unspecified Witt Mass which the newspaper said was sung at the consecration of the cathedral. Print said spreadsheet on good paper and store with music in case .xlsx is an unusable file format in 50 years.
4) Type up a description of your work. How you did it; who told you to, what state the collection was in and its past history as far as you know, and what files and in what format are on the digital media.

I hope I don’t have to perform any of the music you describe any time soon, but the collection itself is irreplaceable and part of diocesan history, and you never know if you have the last surviving copy of a piece.

Burn cd’s. Magnetic media can go bad easily. Burn two copies and keep one offsite.

Seriously, how many boxes?

When can I come take a look / what's the deadline?

Sell the copies on ebay, there are always buyers looking for old mass settings, choral pieces, hymns, etc. Keep copies for the parish archive, if you have one, if not start an archive. A complete list of what you have may help you decide what to do.

IMSLP is the way to go. Those scores will be preserved for anyone who wants them in the future.

A lady recently contacted CMAA with a similar question: what to do with some old vocal scores that her late mother had kept? I asked her to send me photos of the first page of each piece, and then I selected a few for her to send me. I transcribed a couple of them (Easter pieces) for CPDL using Lilypond, since the composers were notable (J. Vincent Higginson and Carl Greith) and I could imagine an EF parish making use of them.

I have pitched a lot of what I've found, kept a few copies here and there, and haven't gone through some of it. To date, my choir has used one thing which I found lying around in a cabinet.

Also, when I was reading this thread title, when I saw "What to do with really old choir..." my mind filled in the blank with "members".

Craig Cramer at Notre Dame sometimes solicits older music for to collect and/or sell.
He has advertised numerous times on our Forum for old choral or organ music.

What about semi-random English language hymn and sacred music octavo scores under copyright from the 1950s-2000? Bonfire?

Music from the transition period (1964-1970) might be worth taking a look at, particularly if it's from one of the older publishers (Fischer, McLaughlin & Reilly). The translations are closer to today's, and they hadn't totally sold out to the folk mass movement yet. Chances are, though, that it's a vanity, and bonfire-worthy.

This was how we acquired our Liber Breviors. The Music Director at the time (more than 20 years ago) was told to throw them out by the pastor, but she (MD) gave them to the Latin Mass community instead, or so I’m told.

I pitched all of the English stuff (~500 scores) and about half of the Latin stuff due to dreckiness. Orchids occupy the freed-up space.

Next up: the Children's choir.

Day One on Church Job
I pulled into the church on my first day and went down to the music room [...] loaded it into four boxes and shipped it out of the church pronto.

Must have been an interesting job description that allowed you to do so, even before saying 'good morning' to your new boss... you didn't get fired in the afternoon, did you?

Thank you, IdeK!! I am obviously not a special collections librarian – just another organist. When I was in such a situation, I asked a college librarian for advice, but evidently the past few years have removed the appropriate order of operations from my brain.