Your mother is a little mistaken. Epoxy does not dissolve styrofoam. Polyester resin (sometimes referred to as "fiberglass resin") dissolves styrofoam. So does spraypaint, and a whole host of solvents.
To dissolve paper, you need something stronger, like concentrated nitric acid. Other things, like water will weaken paper, but won't really destroy it unless you tear it apart.
Epoxy is sold in a couple different categories, based on their intended use.
Most easily obtained is probably epoxy adhesive, which is generally sold in either 2-chambered syringes, or a pair of small bottles. It tends to be thick so that it stays where it is applied, and tends to have a working time of around 5 to 20 minutes after it is mixed.
After that would be epoxy wood finish. It is often not called "epoxy" on the front label. But you can pretty much tell it by being sold in 2 containers that must be mixed. These products are a bit thinner, and most importantly "self-leveling" meaning they try to level out into a flat smooth sheet before they begin to harden. Often this is achieved by a very slow cure time, sometimes as much as 12 hours.
Epoxy resin used in fiberglassing, often available as the more specific "marine grade expoy" is available from boating supply stores. If you need a bunch of epoxy, this tends to be the stuff to get. It has a middle of the road cure time, and middle of the road viscosity.
There are a host of epoxy putties. These have added thickeners so the epoxy will hold it's shape when sculpted. The consistency varies from fresh Play-Doh to firm taffy. This stuff won't work for you, since you'll want to inject your epxoy into the core of your piece.
I think your best and most cost-effective bet would be to buy a couple of the expoxy adhesive syringes. Ideally, if you can find they ones that have a "static mixer" attachment, that'd work well. A static mixer is sorta like a tube with a crazy swizzle stick inside. You screw it onto the end of the double syringe and as you squeeze the 2 components through the tube, it is mixed for you automatically. Using a static mixer, you'd merely need to drill 2 holes in each section. 1 just large enough for the end of the static mixer to fit, and the other can just be a pinhole near the top of the piece to let the air escape. Just keep on pushing the plunger until you see resin coming out of the vent hole. Then apply a piece of masking tape over the holes and balance the piece upright to minimize risk of leakage. Wipe off any leaks with a paper towel before it begins to set.
If you can't find one with a static mixer, you might want to find a separate syringe. I get mine from my local pharmacy. Explain that you want a couple syringes, about 20mL in size, and you don't need a needle. They should be less than a dollar apiece. If you want to try to use a single syringe, you need to wash it out with acetone after each batch before it begins to set. Personally, I wouldn't bother. Squeeze out the unmixed epoxy into a small wax paper cup, mix it with a popsicle stick, then pour/scoop it into the syringe. Then inject it into the hole drilled as described above.
As the epoxy cures, it will heat up. Don't worry about this.