Suzy,
I am 7 years out from treatment, but had many of the same issues.
I think the swallowing difficulty is likely a radiation side effect that just coincidentally appeared around the time of the neck dissection. I had swallowing trouble and trismus both come up around 2 to 3 months after completion of radiation. recovery is a slow, sometimes frustrating process.
The double chin issue is probably a result of the neck dissection. As your body developes new drainage paths to replace the missing lymph nodes, it will improve. As a couple of other posters have said, you might check into physical therapy for massage techniques that can help this along.
I also received a great deal of help with my swallowing issues from a Speech and Swallowing Therapist. After a radiographic swallowing exam, she gave me a group of exercises to perrform, tailored to my specific problem. If your health service has such a therapist available, I would heartily recommend you take advantage.
Of course everyone's experience is going to be different, but many if not most of my issues have resolved over time.
As I write this, I'm eating a bowl of chicken chili, something I wouldn't have even considered 3 years ago.

I hope this info has helped.
I wish you the same good fortune that I have had in your recovery.

Good Health,

Chuck