This just happen to me in 2009 tahoe . Went to get oil change 30 mins later truck started shaking check engine light came on oil pressure dropped call shop told them what was going on they ask to get it towed our cam I make it to to next locations shop .went to other shop was was told it was 3 quarts to much oil in my Tahoe and it threw a code mis fire in 4 … other shop tried to figure how to fix the problem but nothing work was then told my problem is in my engine. So other shop called me and said I must have added the oil that they only put 6 quarts in . So I said why would I put oil in my truck when I just got a oil change . So filed claim with there insurance frist thing the guy tell me was the shop said I add oil to my truck I said no how does that sound. Now still waiting on adjuster to come look at my vehicle.
You shouldn’t have driven it one more foot after you lost oil pressure. Your next step is to hire a good lawyer to negotiate with the insurance company. Don’t try to do it yourself.
While it’s possible that some catastrophic damage was done it is very unlikely. A lifter may have collapsed (no big deal) due to foam getting in it or a spark plug may have gotten fouled from oil. For now you have registered your complaint and if you need your vehicle I suggest you take it to a highly reputable shop that is not involved with the legal situation and get a diagnosis. If the problem is easily and cheaply corrected why not get it done and move on from there. If there is catastrophic damage you can get back to the insurance company and the original shop with the specific$ of the damage if there is solid evidence that the excess oil caused the problem. If you sit still nothing will happen though. The ball seems to be in your court.
It’s possible the shop mistakenly drained something else out besides the engine oil, like the transmission fluid. Good idea to double check for that.