I think the Sharon being recommissioned storyline is the one that suffered the most from the 'jump' forward of a year. Adama was pretty angry with her (for damn good reasons) all second season, but there were hints of his softening. First of all, he sees her as a person with certain rights, even if she was a cylon. Remember when the Pegasus officer tried to rape her? He wasn't as pissed at Tyrol and Helo for saving her from being raped as he was at Cain trying to execute them. Even then, Sharon wasn't just a toaster to be treated any way people wanted.
Also, Adama didn't quite like participating in the hiding of her baby.
In this season, did you notice that he went to her to talk about the situation and get her advice? That was clue #1 how serious the change was. Clue #2, the door to her cell was left open. She wasn't being watched by guards whlie she was talking to Adama, he trusted her enough to sit and drink tea with her while discussing things he would not discuss with his own son. Also, look at the background behind him. A plain, barren cell has been turned into some pretty comfy quarters with nice furniture and everything.
Finally, he allowed his Executive Officer, Karl G. Agathon, to marry her.
There must be a lot of back-story there. With his son not on the ship, losing his edge, his best crewmembers (Tyrol, Starbuck, Dualla) either on the planet or the Pegasus, who else did he have to turn to for heart-to-heart talks? Boomer has obviously filled that role over the last year, and maybe, just maybe, he believes he's learned enough about her from regular talks over the course of that year that he can now trust her on this mission. Notice, he's learned a lot more about cylon capabilities too, as did Starbuck from her talk with Sharon. Another factor was the talk itself. Notice Boomer's comment about having to reconcile herself to her betrayal of her people. Reconciling to a single act isn't exactly easy, but it is not all that hard. Reconciling yourself to a lifetime of betrayal is much harder, and that's maybe what Boomer has had to do...