My understanding of thyroid cancer is that it presents with a distinct lump - not a puffiness. It definitely sounds like you've just been poking too much and everything is a bit swollen (which like I said, I've done to myself today, too).
If you can't keep yourself from poking at it, try this trick: Set a calendar reminder to check your neck in two weeks. If it takes a long time to get doc appointments where you are, set one for three weeks from now. Nothing seriously bad will happen to you in 2-3 weeks, even if it was bizarrely cancer.
The calendar reminder is designed to let you relax - you know you'll follow-up on it, so you don't have to worry about it. 99% of the time when the calendar reminder pops up, the symptom I've been worrying about is either better or gone. And if it's not, you've got the appointment already set.