It is not a standard phrase. It is Stephen King's own invention. Web searches did not turn up any relevant hits except references to King's novel, "Needful Things." There is a standard phrase, "hard-shell Baptist," which I will explain later. A "thicket" is dense shrubbery or vegetation. It is tangled. It too dense to walk through. "Deep-thicket Baptist" suggests someone who is so deep into their religion that they have gotten lost in it and are tangled up in it. "Reverend" is a general title of respect for the leader of a Protestant congregation. ("Baptists" are a kind of Protestant). It implies that the person is an "ordained minister," officially recognized by his or her denomination. Within the Baptists, there are groups with differences in beliefs and practices. "Hard-shell Baptists" refers to a group of Baptists who adhere strictly to conservative Calvinism. Calvinism is opposed to music, dancing, gambling, fine clothes, drinking, and theatre. I don't think "twisting the tails of words" is anything specific to Baptists, I think it is just King's way of saying that anyone who is fanatical about their religion is apt to be intense and emotional when they talk about it.

A thicket is cluster of bushes- so I think it references religious devotion that is perhaps backward and uncultured.

I believe it's a play on the phrase "in the weeds" which means "immersed or entangled in details or complexities" of something. In this case, a thicket would be more dense overgrowth than weeds, so the Reverend is even more immersed in his religion than the normal turn of phrase would imply.

deeply religious