Blick, here are a few other options.
The ideal way to do things is to store them in a safe at home, and use the Bushmaster case for transporting it to and from the range. It works great for that purpose, after all.
The next best way to do it would be to use a case with closed cell foam for storage and transportation. Or, alternatively, buy some raw closed foam and cut it to fit your Bushmaster case. It'll save you some $ on buying a new case. You can also find closed cell foam in some shipping boxes (free in that case), but they can be hard to find.
A very cheap way to "kind of" solve the problem with open cell foam is to cover both pieces of foam with old t-shirts (tucking edges in like bedsheets). Use two shirts for each side of foam, cut off the hem at the bottom and cut off the top of the shirt just under the armpits, then cut the "tube" so that you get one long rectangle. You'd be cutting up four old t-shirts, or, better yet (but not exactly free) you could just get similar material by the yard at a fabric store, probably easier that way. The cloth from the store isn't that expensive, usually like a dollar or two per yard, depending on what kind you get.
The cloth does wick away some oil, but it also tends to hold it in the same location as a big oily spot, unlike open cell foam which just plain sucks it away and dissipates it. This is the method I use on pistol cases, only one old t-shirt needed for those small cases, so it's free.
That's a few things I've tried, from most expensive to cheapest, in that order.