As you stated, you have the wrong type of cable and not enough wires. Go ahead and replace with 6/3 w/g UF (or three #6, one #10 THWN in conduit), run a 50A circuit to the building, and separate the neutral and ground. You still need the ground rods.

As it is, all you have a 20A 120V feeder. If you were still working under pre-2008 codes, you could bond the neutral and ground at the subpanel, install a ground rod (or two) and be OK, but you only have 20A to work with and only every other breaker in your panel will work (unless you jumper the two legs).

Thanks. We're planning on replacing the underground cable (with exactly the size you suggested), but money is an object right now. 100ft of 6/3 w/g UF runs into serious coin. The ground is too rocky and uneven to use conduit. We're in New Hampshire.

I've already jumpered the two legs in the subpanel. All we have are single pole breakers (three 20A; three 15A) and we're careful about only using one tool at a time.

We'll try to win the lottery but until then I'll pick up a ground rod.

The ground rod will do nothing unless you bond the neutral and ground together in the sub panel. The white in your feed will be serving both neutral and grounding functions.

If the ground is rocky, then you'll need to screen the backfill if you use direct burial cable. You don't want those rocks poking holes in the cable jacket.

If your existing buried NM is old enough that it doesn't have a ground, it may be about on its last legs. I've pulled up buried NM with ground where the jacket was literally peeling off. That NM had failed shorted in a lightning storm - a surge finished it off.

If the shed is not attached to the house in any way (structural or any metallic connection except the wiring feeding it power) it is treated as a separate structure.

It will require a grounding electrode system even if fed with a four wire service.
The grounding electrode system is not for personnel safety but for lightening and pole transformer high voltage leakage.
The earth rarely has a low enough impedance to trip even a 15 amp breaker if a 120 V hot line is connected to the earth.

The now banned 'Worm Getter' relied on this.
Two electrodes were driven into the earth a few feet apart, then connected to 120 V.
The electricity irritated the earth worms and they fled to the surface to be collected.

The problem is that people trying to gather the worms often failed to disconnect the electrodes and got electrocuted trying to pick them up.

Sub-panels in separate structures look a lot more like a main panel, since that is what they actually are FOR THAT STRUCTURE.