I have many succulent plants that have leaves with high water retention and stems much the way you described yours. Because they grow so well in my sunny window they start lopping over sideways under their own weight. What I do next is really cool because it creates more plants.....and it's common for many succulents to behave as I describe in the following paragraphs.
So, what I do to help them out is to prune them back by cutting the stem at the halfway point or sometimes even lower depending on how much I want to cut off. The lower portion will begin to sprout new leaves and a stem or sometimes two stems and begin growing again within a couple of months. This new growth comes out of the crook where the topmost leaf attaches to the stem.
Then I take the top portion that I cut off and depending on how long it is I either leave it whole or cut it into two or three portions. Then I strip away the leaves from the lower half of each portion of stem, leaving at least 4 leaves at the top portion of the stem. Then I let them dry for at least 24 hours, but don't let them dry too long or it'll stress them out. This drying out period lets the cut ends and the places where you cut the leaves off heal over, and this will prevent rotting once you plant them. Then once the cut places are dried out, you can plant them in a cactus/succulent soil mixture and keep them watered two or three times a week until they've developed a root system....and they do develope one FAST!
The other way to deal with your plant lopping over is to just let it. This is a natural habit of many stemmed succulents. It'll root wherever its lopped over stem touches moist soil. I let some of my snakey succulents do this because it's interesting....and I get a lot of comments on how weird and neat my indoor succulent gardens look.
One place where you might try to identify your plant is with this web site:
http://www.bryophyllum.com/b/articles/id…
Perhaps it has your plant listed. But so long as you're reasonably sure that your plant is a succulent, you might give my suggestions a try. Most succulents will root this way. Good luck!