Should I try to buy another car?

The issue here is teh TYPE of OIL you are using! NEVER EVER VALVOLINE! It makes SLUDGE and harmful ACIDIC vapors that eat up RUBBER parts and gasket! It is JUNK OIL! USe PENNZOIL PLATINUM that gives a 500,000 miel ENGIEN WARRANTy with prrof of use and change it ONCE A YEAR only! What has HAPPENED is that your idle air control valve and throttle body and even the PCV valve are loaded now with SLDUGE and TAR!1 using AMSOIL POWER FOMAM will remove it EASILY and you can spray into the PCV vlave to clean the RINGS and PISTON TOPS as well! GOOD LUCK!

It sounds like you don't know much about cars. A logical and intelligent would get the car professionally diagnosed to determine the exact problem(s) it has and how much it will cost to repair it. Compare that with what you can sell it for with its current problems and how much additional money you can put toward another car. If you are in California, you won't get much for a vehicle that can't pass emissions. 

Why don't they hook it up to the ECM and try to pull codes or look for open circuits and such? 

the thing will continue do the same thing after you fix the current problems new problems will appear thats what old cars do. eirther get it roadworthy and sell it if you can get a good price for it or do the bear minimum to keep it on the road and put up with things not working  and drive it into the ground.

If you can fix it yourself is what makes an oldcar viable if your paying mechanic to fix it upgrading to newer car will work out cheaper if you choose wisely. The wise chooce will be batshit boring but you will only put fuel in it, 12 oil changes plus tyres and brakes once.

Well honestly it looks like this car is costing you with all of these problems so Why not get a Honda Accord? That is what I have. 

I have the honda accord 2017 lx and it is honestly a great car. My car note is $415 but honestly that is because I bought this in 2018 and it was practically new, one owner and pre certified warranty. Trust me I was in your shoes, I had 04 Malibu that was given to me by my in uncle it was okay just had a dent. But when it started breaking down and giving me problem after problem, that is when I knew that I had to get a newer reliable car.

I don’t know your income but you don’t have to get something brand new, you could go for a 2015-2016 hionda accord and your payment may not be that high. Or get a Hyundai, the sonata is pretty reliable and not too expensive.  I didn’t want a car note at first but t’s worth it and as long as you make your payment it isn’t a problem. 

Time for a new Honda Accord.

......yes, and not another 300C

A car requires maintenance every once in a while.

Between late 2017-December 2017, what maintenance did you have done on the car?

If the only thing you have had to do is get the thermostat replaced and the radiator replaced, that's a pretty good deal for an 11 year old car.

Your check engine light is now on - what is the code for the repair = something little or big?

Based on what you have told us, I wouldn't consider getting rid of the car.

What's your income ? What's your credit score ? Do you have enough cash available to buy a more recent used car ? If not, have you spoken to your bank or credit union about what kind of a car loan you could qualify for ?

No one with any brains buys a used car without first having a licensed mechanic check it over.

You bought a sh*t car and did so stupidly. You are now reaping the consequences of that. Next time, be less stupid and more smart.

I expect you have come across the expression, 'Out of the frying pan and into the fire'. You could end up with the same or more problems with a change of car. Cars cost money to maintain. The check engine light may be a glitch. Find out what it relates to first with a scan of the fault codes. Dirty sensors can send false signals, especially if you have been driving slow around town. I would stick to the car you like.