I was also going to say that alcoholism isn't a single illness, that it has different faces.
There is the stereotypical alcoholic, who drinks a bottle of vodka for breakfast every day, is always drunk and reeks of alcohol.
There is the functional alcoholic, who manages to hold down a job and do most things but who can't function well without alcohol daily and who may or may not have violence issues.
Then there is the binge alcoholic - they may NOT drink every day, and they may be able to go for several days, even weeks, without drinking at all - but when they start, they have no off-switch and get all-out blitzed in a drinking sesh.
And then there are the in-betweens.
So no - they don't all have obvious withdrawal symptoms, and they don't all have a daily need to drink - but they do all lack control when it comes to alcohol intake.
So please don't assume that your H is NOT an alcoholic, because it very much sounds as though he IS, despite his protestations.
Have you been in touch with AlAnon yet? I'm sorry I can't remember if you've said that or not - but if you haven't, please find a group and talk to them. Their input will be even more helpful to you than ours, as they will ALL have been in your position and will ALL be able to tell you the same things that many people on this thread have said too.
Good luck with finding somewhere. 