I have a 2008 Honda Fit with 108,000 Miles. It's been in great condition with nothing major changed/repaired other than tires.

However, I recently hit something hard on the highway. I brought it to the shop and the exhaust was shoved towards my gas tank.

I got the exhaust pipe repaired For $400 because it was undriveable with exhaust pumping towards the engine.
Next, they need to change the exhaust manifold/catalytic convertor. The car burrs like I have a ricer, but it drives.
Their quote is $2,250 for the part and labor.

Do I just keep driving like it is? Or is it time to sell the car and buy another used vehicle?

 

$2250 + $400 for exhaust seems excessive. Got to a few other shops for quotes.

If your Fit runs fine otherwise, I'm not sure there is a compelling reason to sell. At this point there are some maintenance items you need to look at (100k service etc. included in owner's manual).

 

Probably that the most expensive part is the catalytic converter. Look at aftermarket ones, they are way cheaper. I would also put an O2 sensor extension so you will be sure that your car won't throw codes. (From experience, aftermarket catalytic converters are less efficient that OEM ones, and you risk throwing a code after a little while - the O2 sensor extension will prevent that.)