Violent shaking, not sure what to do!

Hi guys, so this has been rattling my brain for almost a week now. I drive full time for Uber and this is my only car so it’s at a point where I need to fix it or sell it.

the problem: about a week ago, while doing Uber, the car started violently shaking when the ICE kicked in. It’s smooths itself out after about 5 seconds, sometimes longer. I received two codes (P0302, P0304). After switching coils and the codes didn’t change, I replaced the spark plugs. Codes went away, the shaking did not. On the 4th of July, I noticed the high coolant temperature light flashed, then came on. Thinking it’s nothing for a 243k mile car, I ignored it as it went away. It would continue to pop up every once in awhile. Then later that afternoon, I noticed white smoke coming from under the hood. When I opened it, I found out that the t-connection that unites the two coolant return hoses to the main hose going back to the reservoir failed. I replaced it, and the issue didn’t go away. I eventually received a new code for P0117, at which I replaced the ECT sensor next to the EGR. No luck. Found out there’s a second ECT sensor and replaced that, code goes away. Overheating light does not. I do recall the shaking going away or being not as noticeable during this time. So I went and bought a used water pump (I don’t have much money since I haven’t been able to drive for almost a week), and the overheating issue went away, but the violent shaking is back.

would it be safe to assume that the water pump is the cause of the violent shaking and if I buy a new one (I don’t want to waste $280 after just spending $100) that it will go away? Or am I better off selling it?

prior to this, the car ran like it was brand new and at 243k it still had a lot of life left, or so I thought.