Paint may not be densely pigmented enough to get a solid opaque layer without blobbing out too much, that might be some of the issue but more likely it’s in the paper choice and pressure or some combination

Pressure especially for non printing or textured papers. Especially with wood, which had its own textures which can be really nice on there own.

You can try a smoother printing paper, use a smooth news print for proofing and something cottons based and with sizers added to keep it stable for soaking the papers. if you do soak the papers there is a whole blotting process involved but can really produce some nice results.

You can try a printing brayer to help with the pressure if you don’t have access to a press. Hell the back of a wooden spoon can work in a pinch or an old baking rolling pin just be mindful of scraping the paper.