Andrea Clary

Do you feel the auto zero needs updated within the 500 second measurement time?  Do you think the instrument is drifting within the 500 seconds?  Is it fully warmed up (1 hr)?
How much drift are you seeing?

Auto zero updates every 2 seconds does not seem likely to give benefit.

If no need to update so frequently, I would just call the auto zero with setting once to force the update then.
Next command to trigger init the 500 second acquisition.

Since you have a number of new changes in behavior, I wonder if one of the power supplies in the 7002 is having some trouble.
The instruction manual on Tek.com for the 7002 has a section on maintenance and running diagnostics.

On the DB9 connector for the digital IO, a 5V supply is available (pin 2).  Look at that on a scope.  Is it steady DC average value?

Our service dept will perform functional check and repair of 7002 still.
https://www.tek.com/en/service-quote

Just expanding on the information provided.

Given that a Keithley SMU is likely to see a few decades of use for perhaps a lot of different tasks, the absence of digital IO for triggering with external equipment is one likely way you will wish you had the 2612B instead of 2614B.

However, there is still the ability to coordinate the SMU action with bus trigger or sending *TRG command to it after arming it for a task.

In the attached PDF, I provide a little more detail on it.