I took the TM out to get weighed, set up as I normally set it up. I also weighed the truck without the axle. I thought I'd post my weights in case someone is curious, but also describe how I calculated tongue weight, etc, in case someone else wants to do this.
I weighed the combo on a CAT scale at a truck stop. These scales have multiple platforms, and, if you can get each axle on a platform by itself, you can weigh each axle. That makes this pretty easy. Cost was $9 per weigh ($18 total).
I packed the TM as I would normally pack - I typically travel without any water in tanks. I didn't worry as much about the gear in the truck as I know I'm way under weight limits in the truck and was looking for the trailer weights (axle, tongue, gross), not the TV weights. But if you are close on your TV, I'd pack as you normally do (including having passengers and such in the vehicle).
TV is a F250 with 5.4L gas engine and nearly full fuel tank.
My TM is a 2007 2619 with A/C, awning, dual 6V batteries, and dual propane bottles. It has the swing tongue.
Weights of the truck, without the TM:
STEER AXLE: 4100 lb
DRIVE AXLE: 2920 lb
GROSS WEIGHT: 7020 lb
Weights of the truck, with the TM:
STEER AXLE: 3880 lb
DRIVE AXLE: 3640 lb
TRAILER AXLE: 2860 lb
GROSS WEIGHT: 10380 lb
First, it's obvious that the front axle of the truck ("steer axle") loses about 220 pounds with the TM on the back. That's what I'd expect without using a weight distribution hitch. For the size of vehicle this is, I'm not too worried about this, but it could be a bigger deal on a different vehicle.
Second, I see I'm not overloading my TM tires - 2860 lbs for the trailer tires. Of course it doesn't tell me if one side (driver's vs. passenger's) of the trailer is heavier, but I think I'm okay there.
To compute tongue weight, I simply find out what the truck weighed before and after hitching up. Note that you can't just look at the rear truck axle weights ("drive axle") because the tongue weight lightens the front axle. But you know the tongue weight has to be the total weight of the combination (10380 lbs in my case) minus the weight of the unloaded truck (7020 lbs - the "gross weight" on the truck-only weigh), and minus the weight of the trailer supported on the trailer axles. Basically, you find the "unaccounted for" weight. Every pound of weight has to go somewhere. You know what the truck weighs empty, and you know what the trailer axle is holding up. Anything else has to be tongue weight.
In my case, 10380 minus 7020 minus 2860 is exactly 500 lbs tongue weight.
That means the trailer weighs 3360 lbs - the trailer axle weight plus the tongue weight.
I know that I'm under the TM axle limits, the truck axle limits (listed on the door post of the truck), my ball's carrying capacity (600 lbs weight carrying according to the ball), and ball's trailer capacity (3,360 lbs is less than the limit on the ball of 6,000 lbs). It's also less than my hitch is rated for, and the gross weight (10,380 lbs) is less than the GCVW specified by Ford.
But it means I can't go over that bridge near one of my campgrounds marked "5 TON MAX". I don't want to find out if the engineer of that bridge was in a gambling mood that day.
Hopefully this helps someone. It has given me some peace of mind about the trailer's weights, and lets me know to watch out for the weight of the water tanks if I use them.
(EDIT: Fixed the weight descriptions, as Rumbleweed noted)
(Edit #2: Dave pointed out I can't subtract. Corrected the wrong numbers!)