Eggy cake with less egg is dry: what next? I'm working to a pretty standard recipe for cake (s.r. flour, butter, sugar, egg, flavours, etc), and it comes out too eggy.

I can usually make a sponge without faff but to do that I use a different recipe. But for this cake, for  non-baking reasons I need to use this recipe with minimal tweaks.

I initially thought it was a cooking time/temperature thing but after a few trials it seems like it just has way too much egg in it. Assuming that it was originally written for smaller eggs, I just reduced the amount of egg and it's now too dry. Not remarkably dry, par for the course for, say, a cherry loaf cake, but this one is supposed to be moister than that.

Is there a way I can add the moisture in a less eggy way? I was thinking of just adding milk or water (say 150ml per replaced egg). Is that too naive? I've heard about tricks involving cornstarch but I can't get my head around them. Would such a trick work?

Current base recipe is (I've tried lots of variants around this and various additional flavour ingredients): 1/2lb each: c. sugar, s.r. flour, butter, 4 eggs. Seived and mixed at each step. Flour in last (which is new to me). 160C (fan) until skewer comes out clean. 4 eggs is v eggy; 3 eggs is dry. Running short of half-chickens to try in between!
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.