Eggy cake with less egg is dry: what next?
Substitute oil for some of the butter.

Here is a trick to  reduce dryness that I just learned from The Perfect Cookie.  Omit a tablespoon or 2 of butter and swap in vegetable oil.  I used safflower last but any mild liquid vegetable oil would do.  

It is awesome for chocolate chip cookies and oatmeal cookies.  I have gone 50/50 butter and oil in the banana bread that I made lately and the texture was good.