Are your neck movements painful and restricted?
Is there any tenderness, swelling?
Any significant past medical or surgical illness?
Are you any medication?
Do you have reflux disease?
A pulsation increasing after the meals need to be considered for gastroesophageal reflux disease. A trial of antacids like Tums or Maalox and acid blocker Prilosec would be helpful (both of them are available over the counter).
Since your gum disease is recent; your stiffness may not be related with it. Many times a gum disease enlarges lymph nodes in the neck. These lymph nodes can irritate the neck muscles which can cause soreness and stiffness with the tight feeling. But as you have this for several weeks; this may be cervical muscle strain (trapezius and sternocleiodmastoid muscle strain). There can be aggravation due to the gum disease and secondary neck lymphadenitis. You can do warm compresses and apply local analgesic ointment. Physical therapy also is helpful, which you would need to learn from a physical therapist.
Please feel free for your follow up questions.
You are very welcome.
Please read this resource for the home measures for the reflux;
Best wishes and kindest regards,