Resetting PID CV so only active on +error

Background: PLC AB 1756-L61 v13 redundant via SRMs

I have a thermal refractory furnace which allows excess air to enter the process to quench the refractory if a temperature exceeds a certain setpoint. The problem is that the PID controller winds up on positive error causing the air to enter and quench, but as the temperature begins to downturn toward the setpoint I still have + error which causes more air to enter so that by the time the setpoint is reached there is enough CV, it takes a long time before the PID CV reaches 0.0 that I am considerable under the setpoint.
Id like to be able to reset the PID to 0.0 CV when the PV is below the SP, but moving a 0.0 to the CV doesn't work as the PID accumulator becomes active the CV begins at whatever value it left off from when the PV is below the SP.
v13 doesn't allow PIDE, and due to this being an older processor with SRM I am limited to a low revision. I need the PID loop to act rather quickly to + error so I can't slow the loop down too much for fear of possibly having the temperature run-away.

Any ideas on how to reset the error and CV, so that the accumulator begins at 0?

This furnace has been in operation for several years. It combusts process fuel to heat a large boiler, but the current logic has caused the temperature to fluctuate enough out of control that operators tend just run everything manually. There are so many mathematical errors that had to be corrected.

In order to force the PID instruction CV to a specific value you need to first set the software manual bit (.SWM), then move the desired value into the set output (.SO) register. You can move a value into .SO and latch .SWM on the same rung, however they will not take effect until the PID instruction is scanned. Then it will be in "software manual" until the .SWM is unlatched and PID scanned.

It appears that PI control is being used and the integrator winds up too much. There should be a problem with the PV going below the SP with proportional only control unless the gain is set very high. The real problem may be caused by a bad value for the integrator gain or dead time. I recommend that the true cause of the problem be found before screwing around with kludges.

Thank you...I've never used the SWM bit before, but I do see that there is an OTE for that bit controlled by the HMI when placing the loop into manual. I moved the OTE into my routine so that I wouldn't a destructive bits set. It didn't work as well as I had hoped. Though it did work logically, Mispeld. Trending refractory temperature PV looked like a sawtooth as the CV kept getting placed to 0.0
I think Peter is right. It may be a poorly set up PID. I need to get better at proper PID setup and tuning. I didn't set this PLC up, but wish I were better at solving PID issues. Maybe I should look more into some of the lessons that you set up once, Peter.