I'm seeing this really late into the discussion - kinda skimmed through it.
I fully agree with everything that
@Scottintexas has been recommending. My only concern is that it seems both engines are overheating simultaneously, correct? Is there some newer safety feature that throws both engines into limp mode if one goes? Aside from that...
As Scott mentioned, how thorough do you think the shop is really going to be? Ask them. Unless they soak these pieces in sulfamic acid (which is not mentioned anywhere in the manual, btw) or replace every part (including the block), there's a very good chance you'll pay for an incomplete job. And then you still may have to deal with a crazy sensor issue, though I have a link in the flush write-up to assist with sorting that out. We have all learned this: no one cares more about your boat than yourself.
This was brought up, too: yes, sand will also be flushed out with my method. The acid breaks up the calcium binder, plus the larger sand particles are pushed out easier by the increased flow current. My belief is the film of salt, calcium, and sand act like a ceramic thermal insulator ...it needs to go.
In regards to your mechanical skills, we all had to start somewhere. That first project is the one that gives you confidence for the next. I tried to write the flush instructions with the beginner in mind (the instructions on my site are even easier to follow) and you'll notice there aren't any fancy/expensive tool requirements. Your engines are a bit different, but we are here to help you through that aspect.
* Once you get the supplies together and the engines assembled (and running), you will be able to flush them well within a weekend's time, no matter your level of expertise.
* That is one of the shortcomings I have found in owning a SeaDoo: the quality/thoroughness of info/help just isn't at the level of support I had gotten used to here - info is very commonly scattered, vague, accidentally incorrect, or incomplete.
I'll admit I have my own selfish reason for you doing the flush, though: I want to see someone else's results using my method. If you've got the time and patience, I say give it a shot.