Car hesitates on takeoff.

You will have to scan for trouble codes to see if a component like the crankshaft postion sensor is failing.
Check the fuel pressure as that is a common issue and you can check it manually with a test gauge using a test port on the fuel rail.

Sometimes failing components won't throw a code. Especially if it is an electrical part and is failing slowly or intermitantly.

Actually it is possible for a failed O2 sensor to cause a backfire. If it reads a lean fuel/air ratio it will dump fuel in the engine which gets ignited in the catylitic converter.
The lack of power is from the rich mixture. It can make a big difference.
Have you noticed if the car is consuming more gas?
With the mileage on the car a new O2 sensor is due for replacement.
It will be worth getting the trouble code scannow as the intermitantly or slowly degrading part has now fallen far enough out of spec that it is throwing a code and that will be worth the 15-30 minutes of labor you might be charged at a repair shop. If you plan to keep the car, readers are getting cheap enough to purchase. Under $100.00. We will always be here to translate codes for you if need be.
I will keep an eye on this post for your reply.

That is correct. I traced it as going back to the Powertrain Control Module. It is part of it only in the sense of high gas levels in the emissions, but not like the upstream O2.
It could be a leak in the exhaust near the O2 sensors or the air injection tube. Irregular fuel pressure could be an issue.
Of course the crankshaft and camshaft positon sneors could have the timing off.
Checking the trouble codes will be well worth it.