[Nathan]

I have always equated 'drowning thoughts' with 'Avoiding dwelling on thoughts that are worrisome/harmful by keeping your mind occupied or focusing elsewhere ' would be as close as I can manage to defining it off hand.

Like someone might put on loud music to 'drown out' thoughts of their relationship problems, or someone might go running to focus their mind on exercise and 'drown out' their issues with a coworker, at least for the time being.

I don't think I've ever used 'drowned sorrows' as a stand-alone expression. Its always been 'drowning/ed his/her/my/your sorrows'. The same goes for 'drowned thoughts'; although I suppose someone more literate than I could work them into some sort of standard sentence.

Disclaimer: I'm usually proven wrong.