Green’s Dictionary of Slang

deeply indebted, i.e. ‘over head and ears in debt’.

1592		Greene Quip for an Upstart Courtier D: The presumptuous asse be drownd in the Mercers booke.
1690		Pretended Visitor Visited n.p.: As occasion served, I reflected indeed on the time when the Church of England was under water, tho' not drowned.
1851		Hants. Advertiser 30 Aug. 6/5: He had taken the things as he believed his employer was drowned. Witness knew that persons technically described themselves as drowned when they became bankrupts.
1961		J. Scarne Complete Guide to Gambling 678: Drowned – said of a heavy loser, ‘I’m drowned’.
1976		Jenkins & Shrake Limo 59: ‘I seen him at the track today. He gimme inside speed, but it was outside closers. I got drowned’ .