I forget who it was that stayed at the Gaylord Palms (:laugh:snicker), but I believe they rented a car in the end.
For sure Disney Transport doesn’t operate outside WDW, and even in WDW only serves Disney owned hotels (plus Swan and Dolphin).
Gaylord Palms probably participates in someone’s shuttle, but you won’t have the same frequency of service as DT/Disney hotels.
A cab will get expensive fast, even if you take a taxi to and from Pop or AoA and then use Disney Transport from there.
Examples: It costs about $20 to go from Epcot to Contemporary, Contemporary to Yacht, and Yacht to DTD with tip.
Check directly with the Gaylord and inquire about their park shuttles to all of the parks, not just WDW.