Sounds like a bad spot in the drywall, maybe caused by water damage, or physical abuse. You're probably best to keep cutting until you find a good section, though you may get away with simply taping and mudding the joint. The tape should hold the section together, so even if the plaster is crumbling it will be held in place. If you opt to just tape and mud, there is no guarantee you won't develop a crack in the future.
If it were me, I would keep cutting (at least to the next joist). If you're lucky the bad section will not extend that far, or at the very least won't extend farther than a full sheet of drywall. You could try to patch it with compound, but it's not likely you'll be happy with the results.
Sounds like you tore off some of the paper on the face of the drywall. This is cosmetic damage and if it wasn't over a seam it doesn't need tape. Scrape the glue (go ahead and damage the wall a little), apply spackle, sand it, (repeat?), paint.
Anywhere with crumpled up paper sticking-out (will not sand flat) needs to be bashed in (using the corner on the butt of a small trowel) and filled; circles with a wet sponge never worked for me. Which is the trick for dealing with the fuzzies where the paper has delaminated and you're looking at a brown spot. Attempt to sponge these types of areas and then apply a skim coat of plaster or it will show in the finish paint. Use a damp sponge, DO NOT saturate the wall, or the entire section may bubble.
An 8-12" trowel makes doing large patches much easier, try not to leave too much at the edges where it meets the old work (this is the most important area to bevel sand). While sanding it, you should begin to feel a nice feather once the old paint starts showing through in spots.
This is kind you want, not a flat trowel with an in-line handle. Those require a bit more experience to use and you still need one of these. If you do buy both, a good tip is to bend the flat trowel a tiny bit, making it ever so slightly concave. This facilitates a nice build-up along the center and gives you an already good taper at the edges.
Best Answer
An alternative that may work for you, depending on the type and height of the existing baseboard, is to replace it with a taller one. Probably need a couple inches above the existing height to make it cover the bottom of the drywall, especially where it has started cracking. If your existing baseboard is 2" - 3" tall then remove it and install ~ 4" - 5" to replace it. If your existing baseboard is already > 4" tall then this may not work so well. Cutting and painting baseboards is a lot of work - but a lot less than patching and painting all that drywall.
If you are going to stain the new baseboards, definitely do that before installation. If you are going to paint them (typically either a gloss or semi-gloss variety of the same color as on the walls or sometimes either plain white or a contrasting color - totally subjective), you can paint them before installation (measure, cut, paint, install) and then touch-up after installation, or you can paint after installation.