•Phone shuts off at around 30%.
•Phone is at 0% when I go to charge it.
To me, that says that the battery is throwing a dendrite. (See
Lithium Battery Dendrite links.) The battery has to be replaced. (It's not Samsung's fault, it's not the battery manufacturer's fault, it's a problem of lithium batteries that the automotive industry has been trying to solve for years.)
And you shouldn't let the battery get any lower than 40%. (Lithium batteries
are not deep discharge batteries.) If you can't recharge it at 40%, and you don't have a power pack to plug into the phone, turn it off (or expect to be replacing batteries). (That's one reason I won't buy a phone with a non-removable battery. I carry a spare, and swap them when the one in the phone gets down to 40%.)