Imagine you had several beers, you are not yet drunk but you feel that your head is heavy, you know, feel good. Is there any specific word for that?
In British English this early level of inebriation is known as being tipsy:
Slightly drunk: tipsy revellers
Obviously its onset will vary across individuals, but in my personal experience the condition of tipsiness prevails in the two-to-four pint range. You're not drunk, but you've got that buzz, you're feeling merry and conversation is at its height of lucidity: the inhibition of sobriety is gone, but nobody's slurring their speech yet.
In addition to tipsy, the word "buzzed" also comes to mind. Might be an American thing.
More generally, you should consult a thesaurus for synonyms of the word "drunk". I'm sure there are plenty of options to choose from.
Light-headed: Unable to think clearly or move steadily, for example during a fever or after drinking alcohol. Synonym: dizzy. e.g. The sun and the wine had made him a little light-headed. ... say Longman Dictionary of Contemporary English.
However I think light-headedness would be the closest to what you mean, and it's hilarious to see they call it lightness in English where the same feeling in another language would be called heaviness!
Another word that might be better to use than light-headedness, is tipsiness.
tipsy: slightly drunk. [LDOCE]. tipsy: The state when you are drinking alchohol in which you are past light headedness but before being drunk. [TheUrbanDictionary]
I think of
Mellow INFORMAL Relaxed and cheerful through being slightly drunk:
everybody got very mellow and slept well
As well as "tipsy", we sometimes say merry in the UK.
Good answers here, older words would include Tipsy, and Mellow, with the latter being not uncommon in Canada, though more so in the 70s and 80s. Mildly inebriated would be technical, and tipsy might not go down well in an all-male group. I wonder what a truly 2000s phrase would be?
For me the description of a heavy head is more of a tired feeling than that of being light-headed or tipsy. I'd probably say "Groggy" would be a stronger fit.
groggy: dazed, weak, or unsteady, especially from illness, intoxication, sleep, or a blow.