The foam board in blue, pink, or green, depending on what area you're from is solid..no beads so it's much easier and less messy to work with. What area you're from also determines the thickness variety. Here we have 1/2", 1", 1 1/2", and 2".
Yes, you want to get the large sheets of pink, blue or green insulating foam...the color doesn't really matter, it is mostly different by name brand is all. These kinds do not get little balls all over, they have very fine grain to them. It's so much easier to works with, can be cut, shaved, sanded and Dremeled perfectly...and is much easier to make details on. We have pink at our stores, in several different thicknesses. If you can find thicker than 1/4 inch, it would work much better for carving stones and for sturdiness. We usually try to get at least 1 1/2 inch to 2 inch thickness...thicker than that is even better, if you can find it. Just remember, this works the same as other foam, though, that you can't spray paint it unless you paint a primer coat on it first...spray paint still eats it.
I've used bead foam for all my projects so I understand your frustration with it. A tip for cutting bead foam - use one of those snap off razor blade knives instead of an exacto or utility knife. They have a longer blade, adjustable blade that is a little thinner than a lot of exacto blades or razor blades. Saw back and forth a little as you cut to instead of just slicing. This will reduce the amount of foam beads that break off. A hot cutting tool works very well too, but can do some serious melting damage until you get the hang of it. There seems to be different densities of it too. I've had some that was soft and spongy and some that was almost hard. The harder stuff carves way easier.
I used white styrofoam packing scraps to make a stone wall and there is indeed different densities. Some bricks carved a little too easy with the foam cutter and some proved more difficult to cut. When doing lettering on tombstones I make sure to start with a new exacto blade and work slow. Still it's easy to make mistakes, and those mistakes is what adds character to the stones.
Good luck with your headstone making. My daughter and I make our own headstones and the tutorials here and ones I have found elsewhere says not to use spray paints because it will eat up the foamboard. I have found a product that looks like stone comes in a variety of colors. I buy it from Ace Hardware. It is called Make it Stone by Krylon. If you want a headstone to look real, I found this works well. You can see some of our headstones on Lil Ghouliette and also on our facebook page https://www.facebook.com/resurrectionvale