Actually going up and down or the Tach is fluctuating?. Tach issue might be computer problem or bad connection or bad sender or the tach itself. Actual engine speed might be transmission shifting erratically or slipping internally. Maybe lock up on the torque converter is intermittently locking and unlocking?
I cut and pasted this from a previous post I made:
Sorry to hear about your issue, but unfortunately I have bad news. I have a 2012 LTZ and that EXACT problem began around 105,000 and the vehicle was still, at the time, under extended GMPP for another 2000 miles. Dealer insisted that the condition is normal and with escalation to Chevrolet Customer Service, the problem was never resolved.
After extensive research, the problem is caused by the pressure switches de-laminating on the transmission control unit. (TECHM). No codes will be set unless the de-lamination is complete. This causes fluttering RPMs while cruising and erratic transmission performance (rough torque converter lockup/release). This results in torque converter damage. At approximately $100 labor rate, it is about 4 hours to replace a TECHM ($400 with programming=~$800 total). To replace a torque converter it pays about 12 hours= ~$1900 total with parts.
You have two options:
Do what I did and ordered a Sonnax pressure switch rebuild kit for the 6T70/6T75 and replace the switches yourself. If you're good at working on cars and a good set of tools you can do this in an afternoon. I recommend a subscription to DIY Alldata as well. The Sonnax instructions are written really well for a intermediate DIYer; and when you're done, the whole repair will only run you about $35. BTW your car has a 1st gen 6T75 transmission.
After completing the repair, the fluttering RPM's were gone but I still had torque converter lockup issues. Long story short I replaced the torque converter myself in my garage (took about a weekend and don't let anyone tell you this cant be done without a lift- it absolutely can). The torque converter was about $185 on Amazon (after core) and a few bucks extra for the input shaft seal. I supported the motor with a Harbor Freight engine support and used a shop crane to lower the transmission from the bottom after removing the sub frame. Its a huge job but can be done. I completed both repairs about a year ago and the car runs like a dream. 147k and still rolling strong.
With that vehicle worth about 6k on a trade, I wouldn't pay a mechanic to do the work. Not worth it in my opinion. Get rid of it before it becomes undrivable.