Not the answer you want to hear, I’m sure, but...
I really do not think there is going to be any simple, straightforward way to accomplish what you want. Damage doesn’t just automatically grow, it’s a side-effect of various specific class features, feats, and items. Which means that any changes to the system are going to require going through each of those specifics to figure out what you need to do with it.
That’s going to mean... something close to rewriting the system. And it’s also going to make using premade modules very difficult, as characters will have to be modified and updated according to the houserules.
It may be easier to simply find a new system that doesn’t have the same problems. It’s going to be very difficult, though, I think to find a rules-heavy fantasy system à la Pathfinder, that has anywhere near the amount of support in terms of premade modules.
Which brings me back to E6. I really think it’s the best solution available here. Scaling characters down in level is annoying, but most likely less work than applying the sorts of systemic changes you’d need to accomplish what you want.
Things I think will not work
A number of the systemic problems in 3.x come from the existing HP inflation relative to previous editions of D&D. For example, most of the Evocation school: those spells frequently deal damage similar to what they did in AD&D, but because everything has higher HP, they’ve lost a lot of effectiveness.
Ultimately, 3.x already has serious problems where debuffs are far more effective than simply damaging things. Increasing HP is only going to make that worse.
This is just awkward, I think. What constitutes a “single source”? For that matter, Sneak Attack is 1.25 damage per level: if you limit it to a max of +level, is a Rogue now expected to multiclass just because his Sneak Attack is too good?
Limitations on multiple attacks
Keeping people to one or two attacks would dramatically reduce the amount of damage dealt, but you’d be massively widening the gap between spellcasters and martial characters, plus you’d make attacking very “swingy” – one attack means it’s very easy to do nothing for the round, and that sucks.
Requested Legend details
I mentioned as a side-comment that I think Legend does a pretty good job avoiding rocket tag, but it’s not really relevant since Legend does not have any premade modules as of this writing, and conversion to Legend, while perhaps not as much work as you might expect, is certainly non-trivial. However, more information was requested, so here’s what I have to say on the matter.
Legend has a lot of very careful mathematics behind it, and the game has had some very thorough “destructive testing” done to it – conscious attempts to break things as a way to find the problems. It also has a very strong commitment to maximizing the significance of player choice. This starts with character building, where the total number of decisions is kept low while the importance of each decision is high so that rather than worrying over small details, you focus on big, important choices. This carries through to gameplay: it is considered a failure of the Legend system for a player to receive no opportunity to respond to a given threat, or for encounters to provide a player no way to contribute.
And most of all, one of the most sacred concepts in Legend is “A = A′,” the idea that a character of a given level should be capable of contributing on an equal footing as an ally, or providing a reasonably equal challenge as a rival, with any other character of the same level. Legend characters can vary a lot, but their level is supposed to tell you how dangerous they are.
I will also give some important details to be aware of coming from 3.x here on this subject: Legend espouses a very steep leveling curve. A character a level higher than you has a big edge. A character two levels higher than you will be tough for your entire party. A character three or four levels higher may, depending on the build, be all but impossible for even your group to fight. By the time you get to a five level difference, there’s almost no combination of lower-level characters who will be able to beat any character five levels above them.
Furthermore, if you are considering Legend, it also “auto-optimizes” quite a bit: Legend characters of a given level are capable of feats comparable to what a fairly well-optimized 3.x character can do, even if you make no particular effort to make them so. If that’s not the style of play you like, you’ll have to stick to lower levels.