For example, in a Newcomb-type problem, suppose I decide to resolve the question of one box or two by flipping a coin? Unless I am supposed to believe that Omega can foretell the results of future coin flips, I think the scenario collapses. Has anyone written anything on LW about responding to Omega by randomizing?

It's not from LW, but here's Scott Aaronson:

(Incidentally, don’t imagine you can wiggle out of this by basing your decision on a coin flip! For suppose the Predictor predicts you’ll open only the first box with probability p. Then he’ll put the $1,000,000 in that box with the same probability p. So your expected payoff is 1,000,000p^2 + 1,001,000p(1-p) + 1,000(1-p)^2 = 1,000,000p + 1,000(1-p), and you’re stuck with the same paradox as before.)