The story is "The Prisoner" by Christopher Anvil, part of the series "The War with the Outs". I read it in the collection The Trouble With Aliens, which I think I found by following a link in another story-identification question on this site... but I can't find that one now.
Your "Tom" is actually named Colonel Gorley. As you read the story, it slowly becomes clear that Colonel Gorley is able to convince human military leaders to implement bad strategy, but he can only do it by speaking to them in person. So while he's fooling people at HQ, the commanders in the field are skeptical about the orders they're receiving, so he has to keep running around. After he leaves, the last person he fooled gradually starts thinking clearly and eventually Gorley has to go back and influence them some more.
I don't think he's a shapeshifter, it's just that his telepathic power includes the ability to make people think they're seeing a human when they're not.
Apart from that, all of the elements match, including Otis's additions.
Here's an excerpt of a message from someone under Gorley's influence:
Office of the Undersecretary for Defense
I have covered the situation for you down at the Opchief's office, and I am sure you must be mistaken about Colonel Gorley. He seems straightforward and solid, and explained the defense setup to me in such a way that for the first time it made sense to me. I can think of no one we might pick who would make a better advisor to the President on military matters.