Yes it will. Usually the top parts are the healthiest, so they're the best part to replant. But even the bottom will grow a new branch if it's still doing well. You can pinch it off at any point & both of the plant parts will continue to grow, but the top will usually do the best.
when I got my anacharis, I just trimmed off pieces of it and tossed it in my tanks. all of it's still there, except the pice I put in my tetra tank, buenos aires tet's must have took care of it.
What I do is trim the tops off my anacharis and throw them into my goldfish tank. Both the top and bottom will continue to grow. This is one of the ways the plant reproduces in the wild.
Sounds like your substrate is either very light weight or very shallow. I find that they stay in place fairly well with heavier and deeper substrates.
As for cutting it. If you leave the bottom half in the tank, it won't continue growing at the top, however if it is healthy the stem may stretch the spacing between the rings of leaves and get slightly longer but more likely other branches will grow off the main stem. If bottom half is not healthy or algae crusted it will just slowly die and rot.
I cut the stem in half, take the bottom out of the gravel and replant the top half. The top half will continue to grow very nicely. The bottom will stop upward growth where you cut it and just branch out all over the place.
I plant the bottoms in a different tank and cut the branches off as they grow to make new plants.