Well, "history repeating" stories with Shinji becoming like Gendo after losing Asuka have certainly been done before.
Generally speaking, I have a hard time seeing how them getting together, even in an unhealthy manner, could lead to things getting even worse than in canon, unless the author gets very creative to stray far off the canonical path, or just goes for an easy "everyone dies" ending (or makes them overly OOC). Let's see:
- Leliel:
a) It could just go as in canon, with Asuka just being a bit less pissed at him surpassing her and more worried as he's gone.
b) With Asuka being happy, Shinji doesn't surpass her, so he doesn't go ahead and it's Asuka being swallowed up.
b.1) Asuka doesn't get out, dies, Shinji goes cray-cray, everyone dies, the end. Rather short one in this case.
b.2) Worried Shinji can't take it, goes after her.
b.2.1) He fails, dies as well, everyone dies, the end.
b.2.2) He/Yui gets them out, things go on.
- Bardiel: Would more than likely just go as in canon. At most Shinji would be even more hesitant to fight EVA-03, if Asuka told him it's Toji (depending if she would even know if she doesn't have a reason to visit Kaji that time).
- Zeruel:
a) Shinji stays in the Bard aftermath for Asuka's sake.
a.1) He's not allowed to fight immediately anyway -> goes pretty much the same as canon
a.2) He's fighting immediately along with Asuka. Still Zeruel is too much anyway.
a.2.1) Zeruel wins, everyone dies, the end.
a.2.2) Asuka gets down first, Shinji's anger triggers berserk mode -> goes pretty much the same as canon
b) Asuka leaves with Shinji.
b.1) They don't come back, Zeruel wins, everyone dies, the end.
b.2) They come back, get into their individual EVAs -> a.2)
b.3) They come back, both get into EVA-02 (cause Rei went in 01?), no berserk, Zeruel wins, everyone dies, the end.
b.4) They come back, both get into EVA-01 (cause Rei went in 02?)
b.4.1) No berserk due to interference, Zeruel wins, everyone dies, the end
b.4.2) Berserk, both are swallowed up
c) Shinji leaves without Asuka - unlikely if the setting is them being codependent on each other, but would probably just be same as canon.
- Zeruel aftermath:
a) Zeruel went as canon: Asuka is more distraught at Shinji being gone for a month (unless love get's him out early or something, but I don't think he really had that much influence over that), but probably wouldn't do anything too drastic as she knows they working on extracting him, so she'd keep up until the first attempt at least.
b) Following Z b.4.2:
b.1) Both get extracted -> goes on as canon.
b.2) They love being together like that and decide to stay inside forever -> Angels win, everyone dies, the end.
b.3) Only one get's extracted, can't go on without the other -> Angels win, everyone dies, the end.
- Arael:
a) Goes as in canon, Shinji being more pissed at not being let out to help her.
b) Shinji gets pissed enough to get out to help her. Whether he "just" rescues her, or shields her suffering the same (sharing their pain) -> better than canon?
c) Her time with Shinji makes the ordeal less painful -> better than canon
- Arael aftermath, this could actually be a point where it could get a bit more depressing:
a) Asuka doesn't get as down as Shinji's there for her (vice versa in case of A b) -> better than canon
b) Asuka still gets depressed and Shinji doesn't know how to handle it.
b.1) Shinji still does his best, but they stay together, even if she drags him down along with her.
b.2) They have a fight and might even split up. Their still remaining longing for the other due to their codependency makes this all the harder on them.
b.2.1) They make up again along the way -> better than canon?
b.2.2) Leads them on pretty much the same path as canon.
b.2.3) Goes as far as Asuka actually managing/going through to kill herself. Shinji's even more down, if he doesn't follow right away, he's on EoE level earlier. Angels/SEELE win, everyone dies, the end.
- Armisael:
a) Goes as canon
b) If Asuka is still happy and ready to go, may alternatively go as RE-TAKE without the "godly intervention". See Aa b.2.3)
- Armisael aftermath: If Asuka's still happy/around, Shinji may take the revelation better, otherwise as canon.
- Kaworu: No matter what, Shinji'd still be all about Asuka, so Kaworu wouldn't have as much affect on him.
a) Asuka is still fine. Shinji'd still be hesitant to kill a humanoid Angel, but he wouldn't be quite as down as having to kill someone who loved him.
b) Asuka is gone.
b.1) Shinji's down enough that he just let's Kaworu go. Angels win, everyone dies, the end
b.2) Shinji's uncaring about anything other than Asuka, kills him with little remorse.
EoE:
a) Asuka is still fine. They fight together from the start.
a.1) They win -> better than canon
a.2) MP's take Shinji/both down, everyone dies, the end
a.3) MP's take Asuka down - well, Shinji already wants to end the world after seeing her dying, so -> same as canon
b) Asuka is gone. Everyone dies/same as canon.
So yeah, seems at most only as bad as canon to me. The only worse variant would be one of them dying, but then you'd be left with a story with only half of the relationship it was supposed to be about.