A Place to Stay for a Little While by Jim Aikin
It was a very unusual household among whose members were a man who patched holes in the world, a shapechanger, and even a talking radio.
It was also a very peaceful domicile until a man who had the ability
to control minds became a resident.
“I can't cope,” Cynthia Lutz said to the radio. ...
Yes, in The Deadly Assassin:
Well, don't you see what he's done? We Time Lords are telepathic. That's simply a brain storage system. He intercepted its forecast that the President was to be assassinated and beamed it into my mind.
The quotation being the Doctor, talking about the Master
While thinking through the potential bromides, the phrase "gay as cheese" popped in my head. One Google search later, and I found Joan Aiken's "As Gay As Cheese", which shows up in Alfred Hitchcock's Witch's Brew, another hardback anthology we had in the house, with this review satisfying me that it's the right one.
Joan Aiken’s “As Gay ...
According to the TARDIS Wikia page on "telepathic contact", yes, in the audio story "The Missing Link":
Telepathic contact was a Time Lord ability to form a telepathic link between multiple minds for high-speed conversations and decision-making. The ability was used by different incarnations of the Doctor (TV: The Three Doctors, The Five Doctors) or two ...
This sounds like it could be Evermore by Alyson Noel. You can read a synopsis on Goodreads...https://www.goodreads.com/book/show/3975774-evermore
It features female central character who moves to a new town as you describe and who can see thoughts by touching a person as you mention.