They are only courted by the Republicans because the party already has a ton of them. The demcrats court them too, but they court everyone - the Demcrats would like their vote but they aren't going to change the party platform to attract them.
The problem is the any realignment of the parties would almost certainly put the democrats in power for at least one election cycle, maybe two. The fiscal conservatives that vote for Democrats because of their fear of the religious right wouldn't know right away if it was safe to vote of the new GOP for a cycle or two.
The way I see it, the only solution is to move over to the Libertarian party and try to do some reshaping. This will of course put the Democrats back in power, but with the current crop of GOP politicians, I really don't see a difference.