Urban detour 3-wheeler is great off-road (puncture repair kit recommended, although we've never had any trouble) and works from birth on. Fine (and comfy) through woods, over fields and on hard-packed sand, but it's not so great on soft sand (what heavy wheeled thing would be?). My sis. used to get the dog to pull her and her DS in his buggy up steep hills...
We used buggies loaned by the National Trust for use on the beach and dunes near Studland in Dorset. They were Mountain Buggy style (3 large wheels, front wheel doesn't swivel, fairly rigid frame) and were excellent to push, even on sand.
Would you be walking on mainly dry sand (stuff at the top of the beach) or the wetter stuff near the water's edge?
I have a Jane 3 wheeler and it is great on the dry stuff but hopeless on wet (just sticks to the wheels and clogs them up).
I'm a bit disappointed with it as on off-road pushchair. I bought it specifically for walking the dog, and I can only use it in dry weather (grids to a halt on wet clay soil). I get a lot of punctures too (we're on our 6th set of inner tubes in just over a year). Could be the way I push it though.