Mine has a V5 receiver hitch/step bumper combination, I think it's rated 10,000 lbs. It has towed a real big low boy with a pintle hitch. I know what you mean about hitch issues, I towed a bunch of old 2X12 laminated stage board from Newport News to Surry County for our Boy Scout camp, total of 19 tons on a friend's car trailer. The first load was probably 7-8 tons, the second was the balance. My 77 with a 390 camper special took all it could climbing the drawbridge crossing the James River, 45 mph at the top! I hit a rough spot near camp at 55mph and bent the step bumper.

 

I brought my F1 from Ohio to Virginia on a U-haul trailer. My 94 F-150 was not "acceptable" as far as suspension and power for U-haul. I needed to get the truck moved so I rented their van as well and my wife drove the F-150 home. One way for the van added about $200.

 

I used to work at a U-Haul place. The trick is to lie. Tell them the tow vehicle is a late 90's 3/4 ton or larger. Dually works best. If I need a U-Haul, I input my '01 Dodge 3500 Dually. Reserve on line, that way it's all locked in wtih what you input, and they have no way of verifying it. tell them you're gonna be towing a Mazda Protoge, or a Focus, or some other compact car. Get the trailer, not the dolly, and it will be better. Use a Subaru Outback for the needs towed vehicle, since it's all wheel drive, and requires a trailer, it's light, and a F-150 should be adequate for U-Hauls needs. Get online, and check various combinations first.

 

Since everyone is offering their experience with towing a vehicle here's mine. With the afore mentioned '48 F-1 panel truck trip I pulled my trailer, a 20 foot tilt bed, a little larger than an U-Haul rig, with my '04 Hemi Dodge dual cab 1500 4X4 Ram. I was told the dual cabbed trucks are basically the same as the 3/4, chassis wise. I had no real problems. I drove 800 miles empty but after I had the truck loaded in Harris burg, PA I drove NE to northern PA to visit a friend and then east to Erie, PA to visit Dick and then home, a total of about 2000 miles. If you've never been in that part of the country it has some of the biggest mountains I've ever been on, except out west. The only problem I had was going up a few of the hills. I topped a couple of them at about 20 mph, getting about 2 mpg. Going down the other side I kept the truck in 2nd. I also had to drive up rural areas that had some impressive hills with very narrow roads.

I have also pulled a '48 F-4 from eastern MN to SE WI, about a 800 mile round trip with the same truck and a friends trailer. I will never do that again. Mostly flat land but I couldn't get the truck over 40 mph without it going squirrelly. I had all the power in the world but my truck wasn't heavy enough to keep the trailer in line when I went around a curve on the interstate. I ended up going all two lane highways all the way home, it was a very long return trip.

 

First off, you don't need to lie to them. Try another U-haul.
I have pulled my 49 F-1, both I used to have a 68 reg cab long bed 2wd 1/2 ton Dodge, and an 84 reg cab short bed 2wd 1/2 ton Dodge. All 3 of them with a 96 1/2 ton 2wd Chevy. Never had a problem renting them.
If you know someone who has rented their trailers a few times, see if they will go with you, knowing the people that work there can make a difference.
But their trailer will handle the weight, and the size, just fine.

As far as the hitch, that's on you. Before you hook up to a trailer you own or use, you again YOU, need to make sure it's leagal. If you have a class II hitch, get another, if you don't have the money, find another way to get the truck home.
Also you need to know the GVWR of you truck, and CGVWR so you know how much weight you can have on it, and how much you can pull. If you good their no worries.

For bobj49f2, if your trailer started to sway, it's not the weight. You can pull 30k on a bumper pull and have it go down the road just fine, with the truck you have. Granted you can't carry close to that weight. But for the sake of argument, say your bumper was off the ground and you had no problems with power.
The reason your trailer didn't pull straight should be you didn't have enough weight on the front of the trailer.
Pull a fullsize truck on a 16ft with no engine trans, you can't get enought weight on the front of the trailer.

 

The weight on the tongue is super important. If you don't have enough, the trailer will start weaving back and forth @ about 40 MPH and you will think its going to flip the truck over. Don't ask how i know this! The trailer has to be built right also. I had a 20 ft trailer that shook the truck and it didn't matter if it was loaded or not. I think the axles were too close to the front. I sold it and bought a 18 ft one. It is smooth as silk, loaded or empty.
I have a 2007 GMC extended cab 1/2 ton, with the trailer tow package, that I tow my 18 ft trailer with. I tow my 49 F-1. I have towed it on all kinds of roads and have never had a problem or felt unsafe. On good roads I run 70 MPH.
To be sure that I had enough weight on the tongue, I set the tongue jack so the foot was about 1"-1 1/2" off the ground when it was hitched to the truck. I then loaded the F-1 on and pulled it forward until the tongue jack just touched the ground. I marked the trailer and put stops on the bed so I would know where to stop in the future. Works like a charm.