The fuel pump is starting to fail.
When the gas tank is full, the level of gas in the tank provides what is called a head pressure. This head pressure assists the weak fuel pump to provide the proper fuel pressure/volume to the fuel injectors.
Then as the level of gas in the gas tank decreases, the head pressure decreases where it no longer assists the weak fuel pump. And the engine starts to run rough.
Fill the tank, and the engine runs fine.