My question is, since we basically live right in between San Diego and Los Angeles, if we can just skip the first leg and get on the later flight at LAX?
As others have noted, Delta expects you to fly your entire ticketed itinerary. If you skip the first segment, your remaining segments are automatically cancelled.
If you skip for final segment, you're still violating the Conditions of Carriage that you technically "agreed" to when you bought your ticket. It's unlikely that Delta would come after you in any way, but people who make a habit of "hidden city" ticketing or "throwaway ticketing" can find themselves in trouble with the airline. In any case, any checked bags would be checked all the way to San Diego, the final city on the itinerary, even if you hoped to end your trip at LAX.
The real solution is to call Delta. Switching you from a nonstop to a connection was not your idea. Delta will work with you to come up with a new itinerary that meets your needs. If the LAX flights work for you, Delta will probably be willing to drop the San Diego segments form your itinerary so that you can
legitimately begin and end your trip at LAX.
Also, even though your ticket is probably nonrefundable, Delta will refund your ticket at your request due to the substantial schedule change. Alaska Airlines operates nonstops between San Diego and Orlando and usually has competitive pricing. You could put an Alaska itinerary on hold and then cancel your Delta flight if that's better for you.
You didn't say if you purchased your ticket directly from Delta (phone or website) or through a
travel agency (such as Expedia, Disney Travel, or a local agency). If you purchased directly, you can call Delta and work out a solution directly. If you bought from a travel agency, you need to work with that agency.
As always under such circumstances, it's best to keep everything friendly, to ask for a supervisor if the first Delta agent can't help, and, if you get nowhere on the first call, to excuse yourself politely and call back so that you'll get a different Delta agent.