As 04_sport_LS states it could be the secondary chain tensioner that possibly could be bad. That would be a pretty easy diagnosis and repair. Just pull the Valve Covers.
As far as the rest of the job. years ago, I was quoted $2500-$3000 from a couple of shops, and even more now. I bought a tool kit from someone on eBay... the Lincoln, LS Jaguar timing chain tool kit. The only things in the kit that worked were the bridge for the cams, the Pin Stop for the crankshaft to keep it from turning, and the shims. The other tools they showed plugging into the cam phaser...on the Lincoln would not engage with anything.
I'm not sure if 04 Sport is talking about the lower long-chain tensioners, but most definitely that is how they work. There's a paw and teeth on the plungers that do not allow them to retract unless Oil Pressure does it somehow internally. If that is true then mine was so tight oil pressure could not release it.
The most difficult part of the job was moving (to the side) the PS and A/C comp only inches... but allowed you to get the bolts on the bracket some Brain Fart designed to hold the pumps. That bracket bolts to the Eng. block and the timing cover. On the Alternator, a bolt and nut were a pain to get to. Also, my Crank Sensor was frozen in place had to destroy it to remove it. (Dremel tool with flexible shaft)
This Link was very helpful...
Detailed repair manual for the 2004 Ford Thunderbird V8-3.9L VIN A.
charm.li