It's all in your Lincoln LS Owners Manual. Your dealer is wrong. Neither Dexron nor Mercon V are correct for the Getrag manual transmission. Lincoln specifies *only* Mercon ATF for the manual transmission. Further, the manual says, "Ensure the correct automatic transmission fluid is used. MERCON and MERCON V are not interchangeable. DO NOT MIX MERCON and MERCON V."
Regarding the vibration and whining, the latter may well be a result of using the wrong fluid in the transmission and/or of having a fluid level that's too low. The vibration may be related to mechanical damage or faulty re-assembly.