It really depends on what you want to do while you're there....and how much you want to spend.
Between Panama City and Destin are some really incredible places...still relatively undeveloped...but pricey....the water is beautiful there....huge sand dunes.
Destin beautiful water, but wall to wall condos....traffic can be horrendous....can be a little pretentious there, and pricey. San Destin Resort is a very nice place...on par with Disney with beach instead of theme parks.
Fort Walton Beach...tacky
Pensacola Beach/Navarre Beach: family oriented, more of a "blue collar" crowd, more laid back atmosphere than Destin, lot of history, Naval Aviation Museum. Miles of completely undeveloped beaches....Gulf Island National Seashore. Island still recovering from Hurricane Ivan, but things are looking good out there. Quite a few new/rebuilt gulf front hotels open now.
Orange Beach/Gulf Shores: Water usually not as pretty as beaches further east due to proximity to Mobile Bay.
Dauphin Island: kind of quirky...water not very pretty
The beaches in Mississippi are not really "Gulf" front....due to offshore islands.
Water does not compare to water in Pensacola and eastward. Still recovering from Katrina.
Weather can very greatly that time of the year...last major snow in Pensacola was in March 93.
IMO, the best time to visit the Gulf Coast is in October, most of the tourists/crowds are gone, weather is usually very nice...low humidity...but gulf is still warm enough for swimming.
Got a particular place in mind, PM me...I'll tell you what I know....lifelong resident of the gulf coast.