Robin Milner, 49, of Sherburn in Elmet, was arrested near the Swan Hotel, Low Street, South Milford at about 20:00 BST on 30 July.
He told York Magistrates' Court he had "participated in a few beers" at a christening earlier in the day and had not realised he had drunk so many.
Milner was given a conditional discharge for six months and ordered to pay Â£105 in costs.
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Magistrates told him they appreciated his guilty plea and the fact he had co-operated with police when arrested.
Milner, who represented himself in court, said he believed he had about four pints.
His cart had scratched a man's car in the car park of the Swan Hotel and the two men were exchanging details when a police officer noticed the defendant appeared to be drunk.

A signalling system at Diss has been damaged, resulting in Greater Anglia services between Norwich and London being cancelled.
Services from Norwich to Cambridge and Norwich to Sheringham have also been suspended or delayed.
The rail company is advising passengers of alternative travel options and of its delay repay policy.
Greater Anglia said disruption on the Norwich to London Liverpool Street line is expected until 21:00 BST.
It said services are unable to depart from Norwich towards Diss and Ipswich and from Diss towards Norwich whilst Network Rail engineers examine the line.
There are currently no replacement bus services available to run between Norwich and Diss, it said.
Greater Anglia has apologised for any inconvenience caused.
All lines were blocks on the Norwich to Cambridge line after lightning struck between Attleborough and Ely.
Disruption is expected until the end of the day.
Greater Anglia said Network Rail staff are on site and are repairing the signalling system however there is currently no estimate for the line re-opening
East Midlands trains services between Norwich and Liverpool Lime St are also affected.

The hosts, who resumed on 23-0 at The Oval needing 230 runs to avoid an innings defeat, lost Rory Burns for 18.
Kumar Sangakkara (84 not out) shared a century third-wicket stand with Arun Harinath (57) before the latter was caught behind off Sean Ervine.
A rain shower saw the players taken off at 16:00 BST, with Surrey on 248-3 and play was called off.
Surrey are now 27 points behind leaders Middlesex with just one match remaining, while Hampshire are six points from safety, but Lancashire and Durham above them have a game in hand.
If Durham and Lancashire pick up a significant haul of points next week, the battle for the final relegation spot could be between Hampshire and Warwickshire, who are separated by eight points with one match left each.
With Hampshire seeking 10 wickets on the final day, Brad Wheal followed up Liam Dawson's early dismissal of Burns by removing Dominic Sibley for 44, caught behind.
However, on a flat pitch Sangakkara remained untroubled, reaching his seventh Championship half-century of the season having hit eight boundaries.
Steven Davies' 57-run unbroken fourth-wicket partnership with the former Sri Lanka batsman saw Surrey creep towards taking the lead in the final session of play before the rain intervened.
Surrey skipper Gareth Batty told BBC Radio London:
"There will be people that will say that after the first four sessions of the game the pitch died and there wasn't a lot (in it) for anybody.
"Fair comment, but you've still got to ride out those pressure situations.
"Hampshire put us on the back foot by going so far ahead, and making it so we were the only team that could be on the wrong end of the result.
"The character shown was great. On the whole, walking away with a draw is not a particularly bad result."
Hampshire captain James Vince told BBC Radio Solent:
"We set the game up nicely for today and the bowlers stuck to it really well and were unlucky not to have a few more wickets.
"On a wicket where it was spinning out of the rough, they have left-handers, if we'd been able to get one or two more in that middle session, I'd have backed us to go on and win it.
"We'll have to see how next week unfolds and then make a plan going into the Durham game (in the final week).
"We're very much still alive, I think."
A man has admitted being drunk in charge of a horse and cart.
[NEXT_CONCEPT]
Lightning strikes on rail routes through East Anglia have sparked travel chaos.
[NEXT_CONCEPT]
Surrey's slim chances of winning the Division One title ended as they drew with relegation-threatened Hampshire.