A recent claim has been that a thawing of the tundra areas would increase methane emissions to the atmosphere. This would seem to put the kibosh on that notion.
If the permafrost were to thaw the rate of evolution of methane would no doubt increase. I do not see the weather being any different than it was 30 years ago, but the permafost was thawing then, so it probably still is. I have seen methane bubbling out of the ground in the yard when there are some puddles to bubble through. But, I also took some of that earth up to chem lab to analyze and it was 100% inert and the water content was pure as we could measure. Since there is no vegetative matter in there, one might assume the methane is evolving very near the surface only.