He needs to just resign and be done with it. That being said, his only crime is infidelity to his wife, the pics and whatever else hes been doing with his twitter account is between consenting adults. Regardless of whether people want to throw stones and call him a pervert he hasn't broken any laws, but being in a position of public office he should hold himself above reproach and failing in that regard he should step down to get his family back in order.
There have been some rumors and talk that some of his twitter-mates were not quite of age. Again, that's only talk at this point, but it certainly makes his resistance to any investigation and his immediate engagement of a lawyer seem a lot more logical.