The solution to both problems seems obvious on reflection. If I was de facto going to give up the negative double anyway, we might as well play the double as Drury. Henceforth, we began to play "Drury Doubles." They are on when the opponents overcall 2 or more, but stay below two of our suit. If they preempt, typically third hand hasn't opened light, so we don't seem to need to avoid problems higher than two of our suit.
Because we gave up negative doubles, we found that we couldn't recover the other major in competitive auctions. As a result, if we are opening light with 4-4 in the majors, we open 1, instead of the more normal 1. (1 caters to partner's being able to respond 1. In practice, that doesn't happen; fouth hand just overcalls, requiring us to use the negative double to get spades back into the auction.) Then, after Drury of any sort (2 or double), 2 by either partner is non-forcing, non-encouraging, and is an attempt to find a 4-4 fit rather than a possible 4-3 fit. 2 denies a normal opening bid, but is ambiguous somewhat about major suit lengths. Opener could be 5-5, 5-4 or 4-4.
If we were left alone, we worked out some methods for Drury followups that catered to this oddity.
This structure assumes one plays fit-showing jump shifts by passed hands. If one plays mini-splinters, change the responses to 2NT to show concentrations of values rather than singletons.
In competition, opener's rebids are the same if available, except that 2NT is natural and invitational, and if 2 is not available, 3-level bids are help-suit game (or slam) tries. 2 is still always pass or correct.
I do not know how much of this is original. The Drury-followups include material borrowed from Allan Falk and others. I imagine other partnerships have evolved this idea similarly.