Is this a continuos beep tone (not beep, beep, beep, etc)? If so, there should be a red light on to indicate an overheat.
I know you just changed your water pump, but did you use the right pump? Did you pull the thermostat and check it and backflush the system from the thermostat housing? If there were pieces of impeller missing, did you find all the pieces? Did you look under the water pump all the way to the water pickups? Many times, the pieces will fall below...clogging the system.
If it is a beep, beep, beep, etc...then you have your motor trimmed too high.
I do not know of any reason a misadjusted shift linkage would be causing this problem.
Mike
Mike