The heat stuck on is 99.9% your dccv. Dual climate control valve. I just had the same thing on the ls i bought a few months ago. It also interferes with the ac and all that. Lots of info about it on here but basically the aftermarket ones are hit and miss longevity, and its recommended to go for OE on this part. However, i just replaced an aftermarket one with another identical one for 40 bucks. Change the red 10a fuse in the box under the hood after u swap the part. Very top-left in the fuse box
These cars are a lot more like european luxury brands, in that maintenance is paramount. Ive had a couple BMWs and theyre the same way. If you keep up on stuff as it comes up, theyll last a long time and are fantastic vehicles. But if somebody cant afford to keep them up on time,and with decent parts, or tends to let maintenance slip unless the car doesnt drive at all, these types of cars arent for them. It seems like the LS is a lot more like a jag than your average Ford in many ways. If there are tons of things needing fixed cause the last owner neglectrd things, it can get add up pretty quick, esp if you are paying for labor