If the float was "sunk" you'd be flooding, not getting insufficient fuel. Needles can stick in the up position from crud or varnish, so that's a possibility but you'll need to open the bowl to find out. A clogged fuel filter will also behave that way on a vehicle with an electric pump, because most pumps have an anti-siphon valve which traps the head pressure. That keeps fuel "bleeding" through the clogged filter after the engine shuts down, so the bowl gradually refills. Some filters have anti-siphon valves as well and will cause strange fuel problems if installed backwards.