Definitions underlined in bold italics, (Abc)* indicating anagram of Abc,
1 Stuck without time to use a straw. (4)
SUCK. StUCK [without] the T (time).
3 Death on march at heart ruined the long run. (8)
MARATHON. Anagram [ruined] of de(ATH ON MAR)*ch [at heart].
9 River Hart turned into wee canal. (7)
URETHRA. URE (river in Yorkshire) and anagram of [turned into] (Hart)*. Not everyday vocabulary, I think. In retrospect I see I could have added "wee" before canal to make it a bit easier.
10 Gasps out rubbish anagram indicator.. (5)
PANTS. Triple definition.
11 Great seafarer, I go East to find the drink. (5,7)
GRAND MARNIER. GRAND (Great) MARINER (seafarer) with the I moving to the right [go East] -> MARNIER. The orange liqueur. "The drink", of course, also meaning the sea.
13 Quiet boy has drink diluted. (6)
SHANDY. SH (quiet) ANDY (boy).
15 A bit of turmeric key to this cocktail with lime and soda. (6)
RICKEY. Hidden [a bit] in turmeRIC KEY.
17 Virtuous frenchwoman with loads of money lost half the quantity of this red. (5,7)
SAINT EMILION. SAINTE (virtuous frenchwoman) MILlION , losing one of the 2 L's (litres, quantity). The famous red wine.
20 Large mixed drink. (5)
LAGER. (Large)* mixed.
21 Still making these drinks. Cheers! (7)
SPIRITS. Double definition. The second one is maybe a little dodgy adding an S to cheer = spirit.
22 Something to freshen the air for an Englishman and his queen. (8)
POMANDER. POM (Englishman) AND ER (Queen of England).
23 No! Not that! (4)
THIS. Cryptic definition.
1 Not going fast like a swallow. (8)
SLUGGISH. A swallow of a drink could be a slug. So "like a swallow" could be SLUGGISH.
2 Claw taps every last abalone at first. (5)
CHELA. C and H (Cold and Hot; taps) and initial letters of [at first] Every Last Abalone.
4 Cyberspace representation of a very active sailor. (6)
AVATAR. A V (very) A (active) TAR (sailor).
5 Hang on in charge? It is a painful experience. (12)
APPENDICITIS. APPEND (hang on) I/C (in charge) IT IS.
6 Women only give a cursory kiss to control their husbands. (3-4)
HEN PECK. HEN (Women only) PECK (give a cursory kiss).
7 Settle down here. (4)
NEST. Cryptic definition. NEST (settle) or (settle down) and (here) the place in which you do it. (Down here) also is somewhere you might find down.
8 Greek character droned on dreadfully about dead tree. (12)
RHODODENDRON. RHO (Greek character) (droned on)* [dreadfully] [about] D (dead). Wishing I could go to Sheringham Park to see the great display there as I did last year.
12 I hear it's fashionable to be an only child in this sleepy state. (8)
HYPNOSIS. HYP sounds like [I hear] HIP (fashionable) and, if you are an only child, you will have.. NO SIS.
14 Setter is endlessly completely up for a soft mixture. (7)
AMALGAM. AM (setter is)[endlessly] ALl (completely) GAMe (up for).
16 Contract is granted to one with a smaller quantity of European drug. (6)
LESSEE. LESS (a smaller amount) E (European) E (Ecstasy; drug).
18 European sort of adulterated coffee. (5)
IRISH. Double definition.
19 Go mad over sweet concoction. (4)