Most do, some don't. It varies widely by state. Few have nearly as broad pardon power as the President does.
In Georgia for instance, the governor has no pardon power. Under the state constitution, before a pardon can even be considered, a convicted felon must serve his entire sentence. Then any pardon can only be granted by the Georgia State Board of Pardons and Paroles.
I sit corrected.
Gregg Abbott needed to get approval from the Texas Board of Pardons and Paroles to pardon Daniel Perry.