Limestone has high porosity and permeability, which means without regular sealing you can’t keep the water out of it and as it’s a retaining wall, you can’t seal the side abutting the dirt.

You can clean and seal it or paint it black .

I would say there are two approaches.

1. Accept it as natural. That’s just what damp limestone does. You could welcome the interesting mosses, algae and ferns which show up.

Or ..... 2. Do away with the irrigated lawn and embrace plantings which don’t need it. We don’t know where you’re situated but if the grass has to be constantly watered maybe it needs rethinking. Perhaps it’s not a suitable, environmentally sustainable or economical landscape solution.

The choice of limestone can turn out to be a happy mistake.

https://www.ugaoo.com/knowledge-center/meet-our-rockstars-no-fail-plants-for-rock-walls/

You’ve created an environment where the limestone stays wet constantly so it’s just doing what limestone does. You could possibly lesson it by excavating a foot behind the wall and fill it with very coarse gravel so moisture isn’t held against the rock but water may still wick up from below. Check that you aren’t overwatering too.

What was done for drainage behind the retaining wall? I have a large retaining wall and a lot of stuff planted above it but there is great drinage inbehind the wall so often even when I water the wall gets only wet from outside and dries quickly. I do agree that limestone has that charteristic and my advice is just learn to love it or drive your self crazy trying to keep it looking new. BTW I planted ivy above mine to hide most of the wall since it is quite high and done in stacking concrete type material so not pretty like a real stone would have been

I agree with Hallett.

I live in a limestone area in a damp climate. Every wall can be a garden.

There are plenty of plants from dry climates which enjoy a limestone wall too.

Thank you all so much for your suggestions and knowledge.
Is anyone aware of a material (like a white wash or acid wash, even a mortar) that we could cover the front of the wall with. Would that work?

Nothing applied to the front will change how the stone acts. It’s a sponge. Love it or fix it from the back. Unfortunately whomever sold you the stone should have set realistic expectations for you.

Thanks for the insight!