Especially on the Ford 3.8L V6 (all years), the head gaskets are very prone to failure because all 3.8L heads are made of aluminum, and aluminum and cast iron engine blocks don't exactly mate up very well. Basic chemistry—two unlike materials expanding at different rates can cause incompatibilities. So the pressure then, literally, is on the factory head gaskets to keep these two materials in line with each other.
A blown head gasket will cause temperature fluctuations at first. Then you'll suddenly get huge plumes of white smoke from your exhaust pipe. This means PULL OVER IMMEDIATELY AND SHUT THE CAR OFF. If you're lucky, you didn't crack or warp the heads or crack the block. The sooner you shut the car off, the better you should be. From this point you can have the car towed and a compression test done, but sometimes you'll absolutely have to have the heads pulled and examined. Usually you'll find the gasket blew out at the #1 or #3 cylinder. If the heads and block are not cracked or warped, then the good news is, you just need to get new gaskets. There is a special 3.8 V6 replacement gasket kit available from Ford that includes new redesigned head gaskets, or you can use Fel-Pro gaskets. Also, it may be a good idea to have your heads shaved down a few thousandths just to be safe. There may be a slight warping of the heads and you definitely don't want to reinstall them in that shape. Machining should be around $50-100 US.
If your heads are cracked or warped beyond repair you'll need some new ones. Now on one hand, they're plentiful and cheap at salvage yards and you can usually inspect them before you buy. But be forewarned that you could end up in the same situation again since this is such a common problem across the board.
We've had excellent luck using new heads from Clearwater Cylinder Head Inc. in Clearwater, FL. The heads are all-new, not remanufactured, from new castings. It is unknown whether this new casting has revised anything from the original design, but we can absolutely attest for their quality as we're running them in one of our cars. CCH stocks heads for all versions of the Ford 3.8L V6. Cost is very reasonable at around $150 US/each, including shipping and the return call tag for the old heads for core.
Clearwater Cylinder Head Inc.
5100 Ulmerton Rd., Suite 2
Clearwater, FL 33760
Toll Free: 800-572-1963
And just so you know, the FWD Continental/Taurus/Sable versions of the 3.8L heads are different from those used on rear-drive vehicles, so be sure to order them for a RWD car.
Now if the new heads work out, excellent...coupled with a pair of good-quality head gaskets, you'll be in great shape for a long time to come. But if you find that that the engine block is cracked, you're in for a whole new set of problems.
You can get remanufactured motors at Pep Boys or other major automotive parts stores for a decent deal. Be sure to check the warranty details before you order. The boneyard is a plentiful source of 3.8L motors...but again, you could end up with the same problem. The yard itself should be able to tell you the condition of the motor before it was yanked. You should try to get a block from the same model year as your car. Generally, for all RWD cars, 1982-83 blocks are the same; 1984-87 are the same; 1988 is unique (internal balance shaft); 1989-93 are the same (naturally aspirated); 1994-2003 are the same (naturally aspirated). The Super Coupe block was used from 1994-2003 on all 3.8L V6 RWD engines, but because of the different block design it will not retrofit using older heads, intakes, etc. For more information about the 3.8L V6, and the parts breakdown by model year, see this website.