Marion, I think it depends on the relationship you have with the facility you'd like to reschedule and the importance of the one you would otherwise have to turn down.
But in a situation like the one you described, I don't think it would hurt to just be very straightforward about the situation, ask if it would be an inconvenience to reschedule -- and maybe offer a price reduction to them for their flexibility --, but stress that you absolutely do honor the commitment and do not want them to reschedule if it will be a problem for them.

I would probably not ask this of a new client, where I had not established a bond or reputation with them, unless it were a sing-in-the-dining-room-while-folks-eat gig that was not a special occasion or a room-to-room strolling time.  Nor would I even broach the subject if I felt my continued employment at that facility was only marginally assured (e.g., with an A D who plays hard-to-get re booking).   But there are facilities where I do general sing-alongs and the AD has a very flexible schedule, and this type of rescheduling, at my request or the AD's,  is not uncommon.  In fact, one thing I tell potential clients up front is that one consideration for which I am willing to use my "sliding fee scale" to accommodate their budget is scheduling flexibility.

It would also depend on how far away the gig is.   When faced with a need to bend the schedule to fit in more gigs (so I can continue to work for the low pay these facilties offer), I'm not hesitant to broach the subject with the ADs if the date in question is a couple of months off.  Once their activity calendar has been printed for the month, it's more problematic.  And there are some gigs where I know the time is fixed  (birthday party last Thursday of the month at 2:30 PM, etc.).

Developing a reputation as a "low-maintenance," reliable service provider, is important, of course.  On the other hand, some opportunities that pop up will probably not pop up again soon if you don't act on them.   E.g., an AD calls to say their regular music person is retiring and they are looking for a replacement;  if you can't do the gig they offer, there's a good chance that whoever does will be hired as a regular and they won't call you again for years.  Or a place that pays really well but doesn't often bring new musicians in wants you to do your special Irish program for their St. Patrick's party.  It may be worth offering to find a replacement musician for the formerly-booked non-special event program or even offer to do that gig gratis if they will reschedule.

I've lost or missed out on some clients because I cannot commit to a rigid, regular schedule such as "first Tuesday of every month at 2 PM"  (because I have too many out-of-town trips), and I've lost some because I've made my schedule too tight, resulting in being late when I got stuck in heavy traffic.   But in over 9 years of doing music for senior facilities full time, I can honestly say I'm sure I have never lost a current or potential client because I asked if they would consider flexing in a situation like the one you describe.   (Some say yes, some say no, a few take me up on my offer of a reduced fee, and there are a few I wouldn't dream of asking.  But it's never been a problem.)

Anyway, now you've got vastly divergent opinions to mull over.  Have we been any help?  *G*

Genie