By asking, he would be revealing his weakness, and would have no future even if cured. All his plans about revenge would be ruined.
By setting things up so that Picard was the closest to the seldom-visited RNZ, he'd get his cure, he'd get it his way (Picard dies), and he wouldn't need to reveal his weakness to anybody (perhaps not even Picard).
Again, I know all that. That wasn't my point. My point was that he should have just dispensed with the silly revenge plan and focused on his own mortality first. But by not asking for Picard directly, he was taking a huge risk that Janeway would sent anyone else other than Picard. Then worrying about revealing his weakness would have been pointless.
Well, no. He was in no hurry in that respect: he spent what looked like at least a day just twiddling his thumbs and doing all sorts of utterly unrelated things, such as toying with Mrs Troi, intimidating his Romulan supporters, having tea and space crumpets with Picard, and hacking into Starfleet datanets. Plus he kept Picard waiting for seventeen hours for unknown reasons!
Clearly, getting Picard's blood was pretty far down his list of priorities.
And that's where he screwed up, because it turned out that it should have been his highest priority since he started dying during his silly revenge plan.
If it so obviously was my lifelong ambition to either run the United States or then bring about its doom, sure I would...
Then it's a good thing that you'd most likely die before your plan came to fruition, if Shinzon was any indication

Incidentally, it's a bit doubtful that Shinzon actually had only a few days to live when he launched his rebellion. Sure, the slowly approaching premature death would motivate him to act, but if the timing was of his choosing, he would have acted sooner. If the timing was not his choosing, then it would be too much of a coincidence that he would only have a few days to live.
Or the more likely scenario, as I keep explaining, was that Shinzon was just a dumb kid who wasn't thinking at all rationally.
Rather, it would seem that his bouts of anger hastened his demise (see all those vein-popping scenes!). Hence, a plan launched at an arbitrary moment would result in him approaching death at an accelerating pace.
But that's just speculation on your part. Nobody in the movie said, "you should probably stop getting so angry, Shinzon, or it might hasten your death." The vein-popping seemed to be the effects of the accelerated aging process, not because he was getting angry.