Starting, all through your thread I have thought what a weak, characterless man this seems. You, on the other hand, have shown real guts in pulling yourself of the floor and facing the world on your own.
Even without his infidelity you strike me as an ill-matched couple. He is not good enough for you. He will drag you down.
You say he turned into a different man when he met OW, but he will always have had the seeds of this bad behaviour within him - they were just waiting for the opportunity to flourish. What may perhaps have seemed an easygoing, relaxed, laid-back attitude was just his weak-willed nature.
He may make his contrition sound plausible, but that is only because like the best liars he convinces himself. He is sorry only for himself. And let's not forget that when he tried to force his way back into your life he didn't even say any of this stuff about realising he loved you - he merely implied you might be in with a chance in a few weeks once he got himself installed! It was only when he realised you weren't in fact going to give him bed and board that he suddenly found he loved you - as opposed to this vague, namby-pamby 'caring' he yaps on about (which doesn't seem to require any effort on his part).
I'm not a dyed-in-the-wool kick-to-the-kerber; I've seen some marriages which have survived infidelity quite well - and one is among the strongest, most loving ones I know. However, even I think this man is bad news. And btw, if you think you are going to be able to help him stop drinking, you are kidding yourself. He can only do that himself, and I consider him too weak-willed to manage it. He'll always turn to drink, I reckon.
However, it is your life and your decision. But as I said up-thread, whatever you do don't let your decision on this be hasty. Especially just because you want a cuddle. Rather than do that, get the train up here to the North East and I'll give you a cuddle.
Honestly, Starting, too much is at risk. You'd need to put a great deal of careful thought into whether you really want to let him into your life, and not decide just because you're going through a bad patch. So many women on this board go on about 'the rollercoaster'; you were riding the crest last week but have surged downwards into a dip because of all the ex-inspired turmoil.
You must not let him into your house or give him money - this is of paramount importance. If he really loves you so much let him man up and support himself, then come back and show you what he's done to prove his sincerity. People are losing their jobs all over the country (especially here in the NE!) but they have to manage somehow. You must not help him - let him help himself. It will do him good, apart from anything else.