Aeropress: Invert it so the plunger is upside down and on your counter. Add the grinds. Pour the water in near to the top, can stir with a spoon if you want. Put the filter (stainless steel mesh) + top cap on. Now, invert (right side up) onto your large cup. Let it sit for a minute. Plunge it about half way down, pull the plunger out, add more water to the top again. Wait a bit, plunge again.
I've seen the inverted method but haven't tried it. What do you think the benefits are, especially in your case where you're still getting some drip after the second fill?