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I have a 79 f150 4wd with a 400, c6, 35" tires, 4:10 gears in a 9" rear end and 3" of lift. Would I be able to safely tow a 19' travel trailer with a max loaded weight of 5700 lbs? Aprox. 480 lb tongue weight.
Damn, the 400 must be a real pig. I towed a 30ft 5th wheel with a 1977 F150 with a 390 camper special engine, 3.25 gear and a C6, it wasn't real happy, but it pulled it. I am much happier with my F350 crew cab DRW with 460, C6 and 3.55 gear, mainly because of the stability and better brakes.
I used to have a similar truck, 79 F-150 with a 400/4spd, but I had 3:50 gears with 33" tires. I used to pull a 20' flatbed with vehicles as big as 1 ton vans on it, and never had a problem. A tranny temp guage would be a good idea with an auto, but I doubt you will have any issues. Trailer brakes would be great too, or at least be extra careful.
I tow a 23' travel trailer with my 79 F-150 with 3:50 gears on 33" tires, but my trailer is a lightweight model. (3700# dry) My trailer has brakes on it, and I use a weight distributing hitch. It trails behind real nice, and the truck still has good power.
You are going to need trailer brakes on your trailer (it should have them already), and a brake controller in your truck.
I hauled a car on a tow dolly once before I had the brake controller, and about fried my front rotors. It took a lot more distance to stop.
-Definately get a trailer brake control put on your truck, and you should be fine.
I can't help much with the transmission part of your question, I have a 4- speed.