What wood are you working with? Unintuitively, harder woods are easier to cut across the grain. Softer woods will tends to crush rather than being cut cleanly. Try both oak and pine, for example.
Also, to achieve good sharpness you typically need to go to high grits when sharpening, which your "old wetstone" probably isn't. You'd want to get to at least 1000 grit, and you'd get noticeable improvement in sharpness all the way to 6000-8000 (I haven't personally used finer compounds, so can't attest to their actual usefulness).
To try out finer grits for sharpening, you can use wet/dry sandpaper at high grits (as high as you can find, big box stores typically carry assorted packs going up to 1000 or more). There are disadvantages to using sandpaper, but the initial cost is very low when compared to good quality stones. Just stick the sandpaper to a flat surface (pane of glass, granite counter top, etc.) and sharpen the chisel.