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Join Date: Mar 2013
Location: Ballston Spa, NY
Should Plan A not work...
I don't mean to get ahead of you, and I don't know of a way to retrieve your information without being able to select "Connect as Media Device" on your touchscreen...but if you plan on getting the same replacement device from Asurion (Sprint Insurance), you could very easily do a motherboard swap between your defective phone and your replacement.
Assuming you file a claim online or at your local Sprint store, just be sure (if asked) to NOT have your replacement device be sent pre-activated. That way, when your motherboards get swapped between the two devices, your original ESN will still be active in its new body. All of your original data (pictures, contacts, SMS, music, etc) will be there. Device works just like it did before, and you'll be sending back the new motherboard with your broken screen. You get all new parts, like your charging port, speakerphone, LCD\TSP, etc but you get to retain all of your information. Asurion doesn't care what ESN they get back, as long as it's the model\brand device they believe it's supposed to be (Optimus G for an Optimus G).
I've been repairing phones at a Sprint store for over 3 years, so if you cannot get your information retrieved and want to try swapping motherboards, please let me know. The swap would take you roughly 45 minutes if you've never done it before, or bring it to a local Sprint store. I'm sure there are also plenty of others on this site who could help as well, but good luck!