Submitting paper proving "X" soon after paper proving "X-epsilon"
This is a great question, and one that I (and probably most mathematicians) feel sympathetic to. In most projects I've worked on, there is some tension concerning the issue of when to cut it off and call it a paper. Sometimes there is a lot of tension. (At least your papers are singly authored. When you are dealing with collaborators, most likely everyone feels that tension, but the forces imparted on the various coauthors are rarely identical and are sometimes all but antithetical.)
The first thing that I want to say is: take a breath and realize that it shouldn't matter too much in the cosmic scheme of things. This kind of thing is exactly why recommendation letters play such a large role in hiring decisions. You definitely want to get letters from people who will attest that you have proved the "Full Cromulence Theorem" rather than just the "Partial Cromulence Theorem". The more eminent your recommenders, the more trust you will be extended and the less pressure you have to "show your hand" at any point in time. But since you already have complete preprints available, in a purely mathematical sense you're done: you've proved the Full Cromulence Theorem and people will take that into account.
The second thing I want to say is: I strongly recommend that you talk to mentors and/or senior people in your field, including your thesis advisor and postdoctoral supervisor.
Okay, but since you asked: I don't think there's one clearly best way to proceed, and I think that you are choosing between things that everyone would regard as reasonable. You could absolutely withdraw the first paper from Journal A and submit the second paper to Journal B, explaining to both journals why you've done so. Then it's a good shot that they will get in touch with the referees of the first paper. Or you could do the same thing but not withdrawing from Journal A: it is not at all your fault that some months later you proved a better result. Finally, you could ask to replace your older submission to Journal A with your newer submission. But it sounds like by doing so you feel that you would be selling yourself short, which I find very reasonable.
To address your specific questions:
Do you think there is any chance that, if the reviewer of paper 1 learns about Theorem 2, they will reject it, saying "this paper is not interesting, as it is just a particular case of Theorem 2"?
Anything can happen, but that would be a distressingly poor thing for the reviewer to say, given that s/he has already had the first paper for many months. I wouldn't worry about it.
Do you think it is likely that the presence of paper 1 will prevent paper 2 from being accepted by Journal B (which has high standards)?
This is a bit more likely, but I think the fact that paper 1 has not yet appeared works for you here. If paper 1 appears in Journal A, then in the minds of some referees and editors it could "put a lower selling price" on the Full Cromulence Theorem: someone can say that the real breakthrough occurred in the proof of the Partial Cromulence Theorem, and even that was only worthy of publication in Journal A, so the new mathematics added in attaining Full Cromulence is not worthy of Journal B. That sounds bad, and it is bad, but look: excellent journals can reject papers by making rather harsh decisions about their value. They do so all the time, and moreover they have to do so. It would be more worth your time to convince the community of the value of the Full Cromulence Theorem than by worrying about what the journals might do.
In fact it could also go the other way: if your community does not know what to make of cromulence theorems for doohickeys, then Journal B, upon receiving an 89 page paper on the topic, could say "The author is working way too hard -- and taking up way too much space -- to prove something of uncertain value. We're not sure that anyone cares about results of this kind." Well, if you publish a partial cromulence theorem in a mid-tier journal, then the community is clamoring for cromulence theorems. Just imagine how much more valuable a full cromulence theorem could be.
So you can't know. Prove the best theorems you can, promote them as best you can, and try not to lose sleep over whether your promotional campaign was the right one.
Should I go as far as to retract paper 1 before submitting paper 2? (However the reviewers of paper 1 probably wouldn't be too happy about this!)
As I said, you certainly could. If you do, you should explain why, and give the editors a chance to enlist the same reviewers for paper 2. But fundamentally it's your work, and you can do what you want with it, including withdrawing it. (By the way, I think it is not clear that the reviewers wouldn't be too happy about it. It is unfortunately possible that they spent very little of the eight months reading over your 73 page paper, in which case they could be quite relieved to have it off their plate.) I would say though that rather than definitively withdrawing the paper, it seems better to explain the situation to the editor and suggest withdrawal. At that point the editor may want to contact the reviewers, and if they are mostly finished the job and like the paper, then that may end up expediting the publication process.
Good luck, and congratulations again on proving that every runcible doohickey is cromulent. Full cromulence! I hope you are proud.
Any other advice about what to do in this situation is welcome!
Well, you are in pretty good company -- the best, in fact -- for proving X soon after proving X minus epsilon. Albert Einstein published the main ideas for the special theory of relativity in his famous paper On the electrodynamics of moving bodies dated June 30, 1905... except that the theory was incomplete, since he had not realized its implications for mass-energy equivalence - the famous equation E=mc2, which nowadays everyone agrees is a crucial part of the theory. So he published that part soon afterwards in a second paper, Does the inertia of a body depend on its energy content?, dated September 27, 1905.
Oh well, progress in science is just messy I guess. Einstein managed to get a job based on his Annus Mirabilis papers, though I seem to recall that it took him a few years.
As for what to do, Pete Clark's answer is just cromulent.
Is the Partial Cromulence Theorem considerably easier to use and/or understand than the Full Cromulence Theorem? Does it cover a large fraction of the potentially useful applications? (Your "X-plus-epsilon" phrasing implies that this is indeed the case.) If so, then I would argue that both papers have value, just to different audiences.
If I don't need the awesomely frobingent power of the FCT in order to demonstrate that my runcible minacule is both bracticating and cromulent, I may well be perfectly delighted to have the PCT available for direct citation, allowing me to use it with far less comment and/or explanation than if I had to invoke the FCT in its entirety.