What options do I have for replacing this sillcock?
You can put a shutoff in the yellow area if there's nothing ELSE feeding off the hidden part of the waterline. If there was a shutoff above the ceiling it should have an access panel, but code violations are a dime a dozen with things like this.

Given the "handle-fall-off" situation I'd suggest overcoming whatever fear of drywall repair may be holding you back and opening up the ceiling to replace the sillcock with a frost-free unit, though I am also of the "and put an inside shutoff on there anyway" camp. Most houses are constructed such that drywall repair is a given for any plumbing or electrical work to get done, so it's a good thing to learn.

However, if you want to check if it's frost free first, and see if you can fix the handle, shut off the water at the main, and unscrew the entire valve-cap-nut assembly to remove the valve from the sillcock - if it's 2" long, it's not frost free - if it's 10-12" long, it presumably is. Then you get to have fun trying whatever you haven't tried yet, and perhaps some things you have, to hold the handle on - thread-locking compound or the like.