I think both were pretty good drinkers, although I think Jack had more experience, given Ennis's early methodist upbringing.
I think Jack's drinking problem is not really tied to an avoidance of his sexual preference and to any sexual confusion -- in fact, I think he seems to be pretty comfortable about it, especially when he is compared to Ennis. IMO, the drinking problem has always been tied to his being unable to get Ennis to commit to a life together. I also think that THAT was the context of Lureen's comment:
"We herded sheep on Brokeback one summer,” said Ennis. He could hardly speak.
“Well, he said it was his place. I thought he meant to get drunk. Drink whiskey up there. He drank a lot.”
(Proulx -- Brokeback Mountain, 1997)
Of course, Brokeback Mountain for Jack was synonymous with Ennis. Drink became the only way for Jack to survive the long stretches without Ennis.
There is a stark beauty in that tense exchange between Lureen and Ennis. It was the point at which Lureen realizes who Ennis was. Her comment about Jack drinking a lot hints at his unhappiness. But we know that his request that his ashes be scattered on BBM did not mean that he wanted to get drunk.
It meant that he wanted to be with Ennis forever.