Have they sent you for physical therapy? It will help get the muscle spasms to settle down. Ever since my car accident in 03, even a small bump causes my neck to tighten up and I get nerve pain and muscle spasms for several weeks up to 2 months. My neck wasn't an issue before that car accident. Now, I know have moderate to severe stenosis at a couple of levels and will need surgery eventually.
You've had surgery. If they do scans and don't see any notable issues, then it's likely that it will just take time to settle down. Physical therapy, massage, acupuncture, etc. should help. Keep with the journal writing. In a few weeks, you can look back in the journal and see how you are slowly improving (or not improving). This will help both yourself and the doctors. Just a word of caution regarding the journal - if you end up having to give a deposition related to the car accident, the other side will ask if you kept any journals. I recently kept a journal for an unrelated issue and the doctor was appreciative that I did as it showed my symptoms are definitely set off by food and I wasn't relying on just my memory. I was denying pain with the symptoms but I'd mentioned pain several times in the journal. So it was a good thing I showed him the journal/chart.
Swallowing - it's very likely related to the neck trauma and it either will settle down, too, or it's a signal of something wrong in the neck. When my spasms get really bad, the first place I notice it now is in the jaw, then swallowing, then I notice that the neck is really tight and spasming.