(Update: Never mind. See the comments.)
Dialetheists like Graham Priest and JC Beall think that some sentences are both true and false, but cleave to classical orthodoxy to the extent of continuing to insist that every sentence is either true or false. (The latter assumption is, in fact, crucial to the derivation of contradictions from the paradoxes.) Thus, Liar sentences get classified as both true and false, and the resulting contradictions are contained by rejecting classical logic in favor of paraconsistent logic. So far, so good.
For obvious reasons, this solution doesn't help with "Curry" sentences, like S1:
S1: If S1 is true, then everything is true.
If S1 gets classified as either (just) true or both true and false, triviality ensues. As such, it had best get classified as (just) false. Of course, this by itself doesn't get around the paradox, since, for familiar reasons, triviality ensues from he mere statement of S1's disquotational truth conditions. That is, however, beside the point for the purposes of this post. The important point is that, for a dialetheist who accepts that every sentence is either true or false but wishes to avoid triviality, the only option for the truth-value of S1 is that it is (just) false.
So far, so good. What, however, about S2?
S2: If S2 is either true or false, then everything is true.