Over the years (especially when he was playing rugby regularly) my DH would sometimes come home much the worse for wear, however he never missed the toilet & pissed on the floor (or if he did, I didn't know because he cleaned it up), he's never been falling-down drunk and never even tipsy in the company of people under the age of 18.
The point is that your relationship is relatively new, it will never be better than it is now.
So he's 'cut down' what does that mean when he gets drunk about once a week? - does he mean that he used to get that drunk twice a week? Three times a week? Him saying that he's not alcohol dependent is no indication of his alcohol use - alcohol dependent people often understate the amount that they drink.
I have a friend who's (alcohol abusing) DH used to not count the tot of brandy that he put in his morning coffee because it was only a 'tot' so didn't count, he also wouldn't count the 2 or 3 pints that he had with his lunch as it was with a meal so didn't count. He used to get through a bottle of brandy a week (in 'tots' in his morning coffee, so not such a 'tot') & God knows how much whiskey, beer, vodka etc. She's amazed at how little she spends every week in the supermarket since he died of an alcohol-related illness.
Until your DP admits to the amount that he's actually drinking and then gets help to deal with it I suggest that you take whatever he says with a pinch of salt - assume that he's drinking double what he tells you he's drinking & be prepared to leave the relationship. Substance dependence will be that person's primary relationship, you will take second place.
I'd also be careful about getting into a car that he's driving - perhaps sit in the back behind him in case he's been drinking & has an accident. That front passenger seat is the most dangerous place to sit in the case of an accident.
I do hope that things work out for you.