I run a flood and drain and have overcome this problem in a couple of ways.
First, try this test to determine the bouyancy of your container....if you fill one of your flood containers with just hydroton and a netcup (no plant), does the cup float when you flood?
If it floats when you flood you might not have enough drain holes in your container. With enough holes your container should not be floating. If you are certain you have enough holes but are still floating your plastic container is too floaty. Find an alternate container. Some plastics are just too floaty, like styrofoam. Sure, you could glue or weld that fucker down but why fight it?
Either drill/melt more holes or switch containers. I use crappy dollar store containers filled with hydroton all of the time with small plants. With the weight of the rocks and enough holes they never float if you have the right plastic.
All of that said, notice I said small plant? A big plant will easily tip a small plastic cup.
I have a solution for that too but it takes a little work.
Ditch the plastic for an actual flower pot, clay pot, ceramic cup, or otherwise cool, heavy, non-plastic container. These pots are heavy and provide a great base for big plants. They won't tip over. They are light proof. If you go to a place like Goodwill you can find any size or shape or level of quirk that you need.......for pennies.
Obviously you need drain holes at the bottom, which can be tricky. If you are handy you can easily cut as many drain holes or slits in the pot as you need. A drill and a diamond hole saw makes this trivial. A set of multiple diameters is less than $10. I do it all of the time. I love flood and drain in awesome porcelain containers. I'm surprised more people don't do it this way.
You can also use one of the super small (1/4" diameter) diamond hole saws to carefully make a bunch of holes around the upper perimeter of the container. This provides solid anchor points if you want to do any LST.
Let me know if you have questions. I do this all of the time.
Best of luck to you!
Flatbush