Double-stuffing will take some creativity here
The good news is that you aren't quite as full as you think, given that your panel accepts double-stuff breakers in the bottom six rows. The bad news is that given that:
- There are no double-stuff AFCIs in HOMeline, and
- We can't move or double-stuff the solar breaker
we'll have to do a bit of rearranging to wedge a power feed to the shed into this panel. In particular, the circuit from space 24 feeding the washer (laundry Small Appliance Branch Circuit) needs to be swapped with any of the circuits from 13, 15, or 17 (front yard outlet, back yard outlet, or furnace). That way, the washer circuit can have a full-sized space for a future AFCI retrofit, while the moved circuit in 24 along with the garage/outdoor outlet circuit in 26 can be replaced with a HOMT2020220 double-stuff breaker to provide the 2 existing 20A circuits along with a 20A multi-wire branch circuit to the shed.
If you take this route, you'll want to run a fat conduit to the shed: 1½" PVC is not out of place here at all, given that a future owner might want to do something more elaborate here, as that can accommodate up to a 125A feeder in the future. If you don't want to faff around with a reducer fitting at the shed end, though, you could run a 1" PVC conduit instead, but that'd make a 2-2-4-6 MHF cable a very tight squeeze, and prevent that future owner from bringing anything larger than the 90A that MHF cable can provide over. Laying a second 1" conduit in the trench for a future communications line while you have the trench open is a good idea, and you'll want to use individual 12AWG THHN wires in the conduit you're using for power purposes instead of trying to stuff a UF cable down it. Finally, your conduit run will need an expansion joint at the house end (for seismic reasons, if nothing else).
Or, you could run power from the backyard GFCI
If you don't mind being limited to a single 20A circuit, one could run power from the LINE side terminals on the existing backyard GFCI to the shed. You'll want to either use a conduit (½" PVC is adequate for this, though) buried 18" down, or UF buried the full 24" with warning tape, though, and you'll also probably need to replace the box that GFCI outlet is in with one that has two holes at the bottom -- a FSS or FDD box in PVC is what you're looking for. Once again, if you go the conduit route, you'll also need to use individual 12AWG THHN wires in the conduit.
Either way, once we reach the shed...
Once we reach the shed, the logic's the same in either case: you'll need either a protective conduit sleeve for the UF cable or a full expansion fitting for the conduit run, then a reducer to 1" if the conduit you run is larger than that, and then your disconnecting means for the shed. Fortunately, a non-fusible air conditioner disconnect is perfectly adequate for this task, and cheap(!) too. From there, you can run 12/2 NM out its back to the light switch and shed GFCI receptacle, which then feed the lights and the rest of the receptacles, respectively, with the receptacles protected by being on the LOAD side of the shed's own GFCI. (You could use a "blank face" GFCI device located at the light switch if you wanted it to be conveniently located without having a receptacle there, by the way.)
(NB for those who have read this far and are curious: the OP's shed only adds 105VA of factored load to their load calculation, or less than half an amp at 240V, so they'd have to be teetering at the absolute brink for the shed to push their load calculations over the edge.)