I had a lesson with Joel Reist (Principal Bass, Nashville Symphony) where he suggested having the thumb rest on 2 strings. That is, if you're playing on the G string the thumb rests on the G and D, etc. It may help to think of the entire LH as one unit for a bit (T-1-2-3). Are you keeping your arm off the top of the bass? I find that the power to press down comes from the weight of the arm and that's not possible if you're resting your forearm on the bass.
Finally, look at your bass's set up and strings, etc. It may be a symptom of a needlessly high action or older strings that have started to stiffen due to metal fatigue. It may be a time for a trip to your luthier to check things out.
You should try the thumbnail on the string(it's half thumbnail and half skin). Check out what Edgar Meyer is doing, I think putting the nail on the string puts the hand in a very strong position.