Yeah, it sounds like you're going to want to basically remove and re-install. The good news is that gutter is actually somewhat flexible, so you'll probably be able to make the adjustments without fully removing all the pieces.
Cleaning the debris from the gutters is definitely a good idea, but it'll accumulate again. If you can get enough slope in the gutters then rainfall might be able to clean some of the debris automatically -- but the shingle sand will always have to be cleaned out manually. Cleaning the gutter before you begin will reduce the weight you have to deal with while working, too.
Yes, you can make an A profile with a middle hill. There are reasons why this is sometimes done but they don't apply on this job. When a gutter is very long or serves a large roof area (ie collects a large volume of water) it can be necessary to divide it into multiple zones. This is the case on my shop building -- it's 51 feet long. With a single slope the gutter would have to hang below the fascia at one end and the collected water would overwhelm the single downspout. Instead that gutter is crowned in the middle and the two ends drop half as much. Where your side A is only 15 feet long this isn't necessary -- but it won't hurt if you want to do it anyway. As I mentioned, gutter is somewhat flexible, so you'll be able to set the center at 1/4 to maybe 3/8 or even 1/2 inch higher than the ends without needing any joint.
Again, because gutter flexes, I think you'll be able to progress from the midpoint of A around the corner and down the B/C sides by unscrewing the hangers some 6-12 feet ahead of where you're working. If you're really lucky you might not even damage the sealant between the corner pieces and the straight sections. If it doesn't work out, complete removal followed by re-install is always an option.
It can be tricky to get the slope of the gutter "just right." Fortunately, it mostly doesn't matter -- just don't make any bellies (low spots). A water level could come in handy. This is a stunningly low-tech, inexpensive, simple tool you can build yourself. It's harder to describe in words than it is to use.. maybe look for a youtube video. You'll need a bucket and a hose. Clear vinyl tubing is great, but if the distance is longer than the amount of clear tubing you're willing to buy, then get a section 10 feet or so and attach it to the end of a garden hose or whatever you already have.
To set up and use the water level:
- Secure one end of the hose in the bucket, and position the bucket so it's about eye level. In this case, that would be somewhere near the level of the gutter.
- Fill the bucket with water to about 3/4 of capacity.
- Fill the hose with water, taking care to purge all air bubbles. Run pressurized water from the end of the hose back into the bucket, or siphon water out of the bucket through the hose.
- Move the end of the hose and watch the level of the water through the clear tube. No matter where you take it, the water level in the tube will stay perfectly level with the water level in the bucket.
- Take care to not allow any water to drain or spill from the system, nor to move the bucket. All of these will cause the reference level to change, and measurements at the new level can't be compared directly with measurements at the old level.
- Fix some kind of measurements to the tube. I like to tape it to a yardstick, but tape or pen marks on the tube work too.
With a bit of math you can work out the relative elevation for any point along the path of the gutters.