What should I say if I am not drunk but I feel that my head is heavy?

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.

Wikipedia also has a nice article about it, in addition to dizziness and presyncope.

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