What you have there is a jade that's not getting enough light, so the branches are stretching out trying to reach the sun. As they get longer they can't hold up the weight anymore and they start drooping.

You're in a very similar situation to this person: http://forums.gardenweb.com/discussions/3016878/not-sure-about-my-jade-plant-is-it-healthy?n=13

The best thing to do would be to cut it back to within a few inches of the soil, put it in brighter light (not direct sun though) and it will leaf out again.

If this seems a little scary, search gardenweb for "chopped jade" and "hacked jade" and you'll see tons of examples. Jades are well-adapted to this process.

If I were you, I would do what Andy said, but keep some of the thicker, more unusually droopy clippings for yourself. The drooping can give you very interesting forms with regards to bonsai culture if you root the cuttings.

This window gets an hour or so of direct morning sun in the spring and some summer but none of that falls directly on the jade. But this window is the brightest in the apartment except for the back porch which gets direct sun almost all day except in summer when the sun is directly overhead. (In my old apartment the jade got some sun through the window. So when I repotted it and then moved is when it started drooping.) Do you think I should move it to the bright back porch or just love it's droopiness? It's also pretty darn heavy so moving it often would be tough. I brought it back inside after a rain storm and it had to weigh well over 50lbs. My son is 40 and he's a piece of cake to pick up and toss around!

This is proof that jades are resilient. I cut a 2.5" leafless stem off a tall clipping and for fun jammed it into soil. It's now totally sprouting leaves! Amazing.

The big guy also has less leaves than before because I just pruned the heck out of it. Took off a ton of bottom leaves and pinched some new leaf buds. I thought maybe that would help send more energy to the trunks.

Once the trunks droop like that, it is difficult to get them 'straight'. Most won't especially if they are so long. Doesn't matter that you got rid of some leaves. If you give plant more light, new growth will be upright. You can see some uprights in the middle of the plant.
The leaves are also far apart - another sign of etoliation. Doesn't matter how many you took off or pinched, new leaves will grow same way (far apart). You will get some branching where the tops were pinched.
If you acclimatize your plant to full sun, the back porch sounds like best location for it.
You have to decide how you want plant to look. Most ppl like them more compact, and not so 'droopy'.

Jades are very easy to grow, they will root from leaves or clipped branches.

If that was my plant, I would take it out of it's pot. I would divide it (I prefer solitary plant, or if a grouping - just few) - plant in separate pots. You can always give extras away... Each plant (or just 2-3 in one pot) will need much less soil and smaller pot, so it will be easier to move. Even in few years as it gets bigger, the smaller pot would be sufficient, but the way it is - big and heavy, it would probably topple over if in smaller pot.

I would prune separated plants extensively, and try for thick, upright trunk.

Of course, this is what I would do...and you may like it as is. So choose the 'style' most attractive to you, and 'go for it'. Your plant looks healthy otherwise, the leaves have good color.

Just couple of examples:

The plant below was very 'droopy' and branches had some frost damage. I pruned off many leaves and damaged branches. It will be pruned further later on. See how much more upright and compact the new growth is.

jades before and after - click here