Hi musicmaker .....
As a retired musician, I can understand your frustration with no longer being able to hear 100% in both ears.
If it is any consolation, your brain will eventually learn to compensate for the loss of accurate input and block out the poor quality sounds.
As you can read in my signature, I have had two AN surgeries on the same side. Following the first surgery, I retained 20% of my hearing and still maintained 100% speech discrimination, so I was fitted with a traditional air-conducted hearing aid. The horrifying thing was waking up to discover my AN ear was about a 1/4 tone off from my remaining good ear. This resulted in all music, but especially piano music, sounding horribly out of tune. It took several months, but eventually my brain blocked out the wrong pitch input. During that time, I found it very disconcerting to try to sing, especially in a choral group when I could not hear myself at the same time as others. Side note: prior to my AN diagnosis I had perfect pitch. Following the first surgery and ever since, I no longer can discern a perfect pitch.
Since my second surgery, I am totally deaf on the AN side, but now wear a bone-anchored hearing device, which has been wonderful for hearing sounds coming from my deaf side, but it will never be the same as complete, natural hearing again.
You mentioned the sensitivity to loud sounds. This is called hyperacussis. In single sided hearing impaired people, it is usually a result of the brain overcompensating and perceiving sounds way louder than you should. I have this as well ..... especially loud music, crinkling bags, clanking dishes/silverware, etc. I had a custom-fit musicians' earplug made with a filter in it that does a good job of filtering out the annoying pitches to use in my "good" ear. I use it for most concerts where I know it will be loud (like brass instruments, loud monitors, etc.). I also use it to filter out road noise when traveling in a car.
I doubt that you are singing off key, it just feels that way because of the loss of accurate input. Have someone you trust, listen to you and confirm that you are still on pitch. Also, remember it takes the brain longer to catch up to loss of input than one would think.
Hope that helps.