A month ago, I had a neck thread lift! Not happy.

"Thread lifts" are beginning to regain popularity because of the increased safety and more predictable results than in the past. That being said, they will never replace or achieve the results of a surgical face or neck lift.  To compare a thread lift to a true neck lift of any kind would be unfair to patients and both procedures. Although patients may get excited to hear its done under local anesthesia,  those who have become part of my practice know that I perform all of my facial plastic surgeries (including facelifts) under local anesthesia as well, which creates a level playing field when it comes to that part of it.  In the right patients, I have found that the results are worth the time, effort and resources. Thread lift procedures are a minimally invasive way of tightening the skin of the face or neck.
The only way to know for sure what your best treatment options are and get an unbiased opinion will be to see a board certified facial plastic surgeon who performs a wide variety of neck tightening procedures (surgical and nonsurgical).  If there is excess skin and/or fat, then you are correct that this will likely need a surgical procedure. Nonsurgical treatments are focused on repairing skin and building up it's collagen (tightness). These include radio-frequency, ultrasound, and laser.  If you decide to start with surgery, it's important to understand that surgery will allow for tightening of muscles and removal of excess skin. However, the "age" of the remaining skin (i.e. collagen quality) is unchanged. What I find is that sometimes combining approaches can be necessary to achieve the goal. It's not too early to be evaluated as to whether or not a neck lift will be helpful in your case.  Swelling from thread lifts is minimal and should be completely resolved by this point. 

Threat lift "neck lift" is never going to give the excellent results that a neck lift does. In an aging neck, the muscle becomes loose, there is excess of fat and redundant skin. Thread lift by just pulling the skin, just camouflages the aging process. In the other hand, neck lift plastic surgery by removing and tightening the loose neck muscle, removing the excess of neck fat with liposuction and redrawing the neck skin and excising the redundancy does deliver a rejuvenated neck. Patients interested in full neck rejuvenation  should make a consultation with a Board Certified Plastic Surgeon specialized in plastic surgery procedures of the face and neck 

A face and/or neck lift may be possible at this point.  You would require an evaluation to assess how the healing from your prior procedure has progressed, and to determine which approach would be best to address your concerns.  During the surgery the "threads" can be removed if needed.  Best regards.