Unless you are referring to a very specific "chagrin", the plural "drown his sorrows" would be more natural and, in my opinion, "in alcohol" is superfluous. The expression "to drown one's sorrows" is already a well-known metaphor for using alcohol to escape from distress.
The expression that we usually use is 'to drown one's sorrows', so just pluralize the word sorrows. It is not even necessary to say 'in alcohol': -He would like to drown his sorrows.
Having thought about it, I realise that it would also be possible to use your original suggestion: 'He would like to drown his sorrow in alcohol,' but there is a subtle difference between them.
'To drown one's sorrows' means to drink to forget one's problems, or to get over an upset of some kind (quite possibly something of little importance). It is very clichéd.
'To drown one's sorrow in alcohol' is much more literal (and literary). It suggests that the person feels deep and genuine sorrow, and really does intend to use alcohol to eradicate it from their mind, if only temporarily.
That is how I would differentiate between the two, anyway.