I’m not a fan of oil cakes, myself, but many people prefer them. They do have a “super-moist” feel about them—to me, it’s spongy and sticky. But of all the chocolate cakes I’ve made, two people picked the German Chocolate “cake part,” which is an oil cake, to be their favorite of all, so oil cakes are a definite way to go.
Another possibility is to syrup your butter cakes. Again, I’m not a fan, but many are, and, if the weight of the sugar in the syrup is equal to the weight of the sugar in the cake, it will moisten, but not sweeten, the cake. Several of Rose’s recipes call for syrups—the lemon poppyseed bundt cake from RHC comes immediately to mind—but you can syrup any butter cake. You can also flavor the syrup with liquers and such if you want to.
In cakes that have more fiber, like Rose’s Cordon Rose Banana Cake, you can add additional “weighty” ingredients like sour cream—she gives a starting point but, in the footnote, says you can use up to (I think it’s 4x as much) for extra moistness, becuase the fiber in the bananas can support it.
Of rose’s cakes, I’ve found the Golden Luxury Butter Cake to be the moistest and fluffiest. It uses white chocolate for part of the sugar (and fat), as well as all egg yolks. It’s my favorite yellow cake. I think that buttermilk is supposed to produce a very moist cake, so you might want to try the TCB buttermilk country cake and see if you like it. It also uses all yolks, but you can sub 2 eggs for 4 yolks if you prefer (I like all yolks, myself)—Rose does this when she uses this cake as the base for the RHC spice cake.
Using mayo would be rather akin to using oil, I would think.
On the whole, though, I’d think (1) oil cakes or (2) syruping would be your best bet.
Something interesting about oil: Rose has two banana cakes: The TCB Cordon Rose Banana Cake and the RHC Banana Refrigerator Cake. These cakes are identical, except that Cordon Rose is all butter and Banana Refrigerator Cake uses all oil. Later, Rose modified the Cordon Rose by suggesting bakers substitute 2T of oil for 2T of butter, to make the cake moister. I personally love the original, but this gives you an idea for the same cake made 3 ways—at 3 levels of “moistness.” I personally love the original Cordon Rose with the maximum sour cream!!!
I hope some of that is helpful!!!