Want to run a new Electrical cable to new shed
Can you abandon the old wire in place? Sure. Well, technically I think I recall reading something in NEC about removing permanently-abandoned wiring where feasible -- let's just say it's infeasible to remove that direct-bury cable and move on.
Yes, if you up-size to a 20 A breaker you'll have to replace the 14 gauge wiring in the shed with 12 gauge.
So far as circuits and wiring from the house to the shed goes you have options. A detached building is allowed to have a single branch circuit or a feeder circuit. Choosing the feeder option entails installing a subpanel in the shed and a grounding electrode system too (it's not terrible, just a little more work and cost than you had in mind).
&quot;A single branch circuit&quot; actually means more than a person would expect. If you wire that as a &quot;multi-wire branch circuit&quot; (MWBC) then you can have two 20A, 120 volt lines going out there while still calling it one circuit.
There is another trick for gaining an extra circuit: Add some kind of outdoor lighting on that shed. You can have a switch in the house that controls the power to that lighting circuit. You can also have a switch in the shed that controls the light. And if there are a few outlets on the circuit too, that's ok.. To complete the idea, just leave the switch in the house turned on all the time so that the outlets are always live and ready for use.
The new wiring to the shed needs to be UF-type cable which can be direct-buried or pulled (wrestled) through a grossly-oversized conduit, or else conduit with wet-rated loose conductors like THWN. Don't use NM (&quot;Romex&quot;) cable for the underground portion. The conduit wiring method is unfamiliar to many people, but it's worth consideration.