My LFS sells a kind of adhesive that supposedly works. it comes in a tube and you work it with your hands and then apply it to whatever youre building and let it set up. They use it on the corals in the marine tank and the slate as well as bog wood in the freshwater tanks. Check with yours and I'm sure they can point you in the right direction. Hope this helps.
That would be Epoxy..... Which needs a something to get a hold of. It is not sticky, but hardens into whatever you want to attach. Will not work with smooth surfaces.
Go with the Weldon, it will fuse the plexi and make it like one piece.
Weld-on is what you use when you make acrylic fish tanks, sumps and over flows. I use it all the time. Easy to work with. I do not believe that Home Depot or Lowes carries it. You may need to go to a plastic distribitor or order it.