Any tips to make my sourdough bread moist?

When I add leftover grains or beans to the sponge (not starter) the bread comes out very moist. Even if I just use half a cup of rice for 4 large loaves.

But this assumes you make a sponge from your starter the night before baking. How you make your sponge, runny like water, or thick like almost dry cement will have a large effect on the flavour and moisture content of the finished bread (moist sponge for mild, thick sponge for strong sour taste).

If you are cooking directly from your starter, try making up a sponge the night before, then bake with that (feeding your starter separately).

Just a clarification on the terms I'm using here. Starter: the flour/water mush that your yeast lives in. Sponge: made from starter the night before (or sometimes several days before) baking. Often made with just flour, water and starter, but sometimes including leftovers or other elements that would benefit from a bit of fermenting before going in the bread. Sponge was traditionally runnier than a starter to get the yeast even more active. I think because most people these days keep a runny starter (you spoon won't stand up in it) that they often forgo the sponge stage - but I find it's really helpful for controlling flavour and texture of the finished loaf.

Potato flour works wonders . . . replace a couple tablespoons of flour with it, best done by weight not volume. If you can't get potato flour, use potato flakes (instant mashed potatoes without any seasoning or flavor packet ingredients). ;o)

I'd be curious to see your recipe. I make sourdough bread with a pretty high hydration percentage (70%, or 700 grams water to 1,000 grams flour) coupled with a slow rise. This makes exceptionally moist and flavorful sourdough bread. It's not loaf bread--it's shaped in a boule or round--but it's some of the best bread I've ever had. I use Tartine's recipe and would highly recommend giving it a shot.

Here are some tips for troubleshooting bread from King Arthur Flour,
http://www.kingarthurflour.com/tips/bread-troubleshooting.html
There could be a number of things that are happening to make your bread dry. I don't think adding flax seed would necessarily help. Check out the tips, KAF has knowledgeable and reputable bakers providing these tips.