I have not seen your x-rays or mouth, so I cannot specifically diagnose anything; nor would I accross the internet.
Although, neck pain and local lymphadenitis can be tell-tale effects of an impacted tooth. If you have bumps around the neck area, those are lymph nodes 'dealing' with a localized infection which is normal unless they do not disappear after a long period of time. The actual neck pain can be caused by an impacted tooth putting preasure on the other teeth, essentially giving them no place to run. This could force a crowding effect of the teeth which may result in neck/ jaw pain. Also, the impacted tooth may have impinged the gums and/or created a food trap that facilitates tooth decay and/or gum disease which would cause lymphadenitis/localized pain. There is a myriad of other signs, but this is more in relation to your situation. And, this can also be attributed to erupted teeth as well, not only impacted teeth.
I would suggest that you listen to your provider and get them removed; especially, if it is the underlying cause of your neck pain. Moreover, considering your intolerance for pain, try to seek treament from an actual oral surgeon rather than a general dentist if your insurance covers it or you can afford it. People who see OSs usually recover more quickly and with less complications, simply because that is part of their specialty.
For the mean time, floss and brush around your wisdom teeth thoroughly. If slight bleeding occurs, it may be because there was food stuck or impinged gums that caused gingivitis. You may also try to garggle with warm saltwater, just tap water and salt will work - bad taste, but it can help alleviate some problems. Teeth involve the whole body too, so take care of them
Hopefully I helped you a little. Take care.