Really, abdication has nothing in it for the King. His successor is already safely abroad and in place to return to rally the Royalists if need be; he might as well follow the path he did IOTL, which was to negotiate in bad faith, divide his enemies as he had done several times before, and prepare to be martyred if that was neccesary.
Charles' 'conscience' mattered more to him than anything else; it's what caused the Wars of the Three Kingdoms in the first place. If Charles wasn't willing to sign the Covenant and wasn't willing to concede any power to Parliament, he certainly wouldn't be willing to abdicate, which would after all be an abrogation of his coronation oath.
Speaking of the Covenant, it's worth remembering that in 1649 Charles isn't just King of England and Ireland, but also Scotland. This makes Abdication even less attractive for everyone concerned, as it means that there is still a Kingdom for the Monarch to flee to and raise his standard, should the need arise (it would involve the usual stand-off over signing the covenant, but it's not as if Charles hasn't strung the Scots along twice already).
This leaves enforced abdication, and this is something that Charles was worried about IOTL. His eldest sons are both non-starters as successors; Parliament would have been mad to have crowned either of them, given that both were die-hard Royalists who would only return at the head of an army. That leaves Henry, the Duke of Gloucester and Charles' youngest; he was under Parliament's control, had been given parliamentary tutors and was regarded as a good protestant child. He's an unlikely alternative king though; as the Princess Elizabeth recorded during their last meeting with their father,;
"Taking my brother Gloucester on his knee, (Charles) said, 'Sweetheart, now they will cut off thy father's head.' And Gloucester looking very intently upon him, he said again, "Heed, my child, what I say: they will cut off my head and perhaps make thee a king. But mark what I say. Thou must not be a king as long as thy brothers Charles and James do live; for they will cut off your brothers' heads when they can catch them, and cut off thy head too at the last, and therefore I charge you, do not be made a king by them.' At which my brother sighed deeply, and made answer: 'I will be torn in pieces first!' And these words, coming so unexpectedly from so young a child, rejoiced my father exceedingly."
All of this leaves the "Hanoverian" option of finding a new, more suitable King. Probably the best candidate for this was the Elector Palatine, who had been mooching aorund London for most of the mid-1640s in the case of just such an eventuality; as of 1649 though, he'd just been restored to many of his former lands and might not be as willing to take the throne, even if Parliament had wanted him (which was pretty dubious).