I always noticed that overcooked cakes shrink too much, so cook them only until there is no wet on a toothpick inserted in the center. Don't go beyond that amount of time, even if your recipe says to cook longer. Your oven might be different. You might also try not greasing the sides, just the bottom. Let them cool ten minutes in the pan, then cut around with a thin knife before unmolding the cake onto the rack.
Why do you want the cake to be exactly the same size as the board? If you're stacking layers to make tiers of a wedding cake, the cake boards can be even an inch smaller than the cake, so get them smaller. But if it's just to have something to put the cake on between teh cake and the serving dish, you can still get the board a little smaller.
I've found square cake boards in london, and i imagine you could order them online somewhere. You don;t say where you live, but i'm assuming the states?
I've also found thick cake boards (maybe a quarter inch, maybe a little less) and i believe they were some kind of thick and strong styrofoam covered in foil or something.