DRG DSCNT FM FL 350 FOR IFR ARR TO NEW ORLEANS, FLT CREW NOTED GREEN & YELLOW RTRNS ON WX RADAR WITH SOME ISOLATED RED CELLS, LEFT & RGT OF INTENDED FLT PATH. BFR ENTERING CLOUDS AT 30,000', CAPT SLCTD CONTINUOUS ENG IGNITION & ACTIVATED ENG ANTI-ICE SYS. CREW SLCTD ROUTE BET 2 CELLS, DISPLAYED AS RED ON WX RADAR. HVY RAIN, HAIL & TURBC WERE ENCTRD. AT ABT16,500', BOTH ENGS FLAMED OUT. APU WAS STARTED & AC ELEC PWR WAS RESTORED WHILE DSCNDG THRU ABT 10,600'. ATMTS TO WIND- MILL RESTART WERE UNSUCCESSFUL. BOTH ENGS LIT-OFF BY USING STARTERS, BUT NEITHER WOULD ACCELERATE TO IDLE; ADVNG THRUST LEVERS INCREASED EGT BYD LIMITS. ENGS WERE SHUT DOWN TO AVOID CATASTROPHIC FAILURE. EMERG LNDG WAS MADE ON LEVEE W/O FURTHER DMG TO ACFT. INV REVEALED ACFT ENCTRD LVL 4 TSTM, BUT ENGS FLAMED OUT, THOUGH THEY HAD MET FAA SPECS FOR WATER INGESTION. ACFT HAD MINOR HAIL DMG; #2 ENG WAS DMGD FM OVERTEMP. AFTER INCIDENT, OMB 88-5 & AD 6-14-88 ISSUED TO RQR MINRPM OF 45% & TO RESTR USE OF AUTOTHRUST IN MOD/HVY PRECIP; ENG MOD WAS PROVIDED FOR INCREASED CPTY OF WTR INGESTION.