Agree - that to me sounds like classic behavior for a bad sensor. The part is cheap - $15 for a decent aftermarket, $35 for a Ford part. Replacement is easy - $35 labor tops.
The other possibility I could see is that the there’s air in the coolant system. I’ve heard (no idea if it is true) that these were not easy to bleed all the air out of, and I’ve even heard that the LS had a bleeder for the cylinder head near the master cylinder to help get the air out. I have NO idea if that is true - I’ve never heard of it before or on any other car.