Oops, disregard my earlier post. We have an apples & oranges situation here. You asked about replacing the 'sole plate', and I jumped in half-cocked with a 'sill plate' replacement. So sorry.
On to the problem at hand. Are the walls plaster or drywall? If the latter, I would say cut away enough wallboard to adequately get your sawzall to cut the stud to sole plate nails. Then cut the rotted sole at the appropriate location and replace with new. Of course you have the drywall repair to complete then.