The TH350 in my 69 seems to be shifting too quickly. When I take off from a dead stop under normal acceleration, it shifts from first to second at what seems to be the correct speed. It then shifts from second to third immediately after shifting from first to second. Is this normal? I would expect a longer delay in between the two shifts. Could a simple band adjustment take care of this?

No. The shifts are controlled by the governor mainly, and the modulator to a minor extent. They act on main line pressure on the shift valves to make the shift points.