I believe He does. At one point in every person's life, there will be a chance to get saved, where He draws them. If they refuse too long, He doesn't draw anymore; hence "no man can come unless he is drawn" (paraphrase). A person can't just decide to be saved. There has to be a drawing. But all men will get drawn. It doesn't mean only some can get saved. It means that you can't just decide to at anytime.
Where it gets hairy is those who never hear the gospel. I believe that God, in His foreknowledge, knows who would respond positively, and makes sure that those people get a chance.
But, to use your words, I resisted Him. For almost 18 years. So, there goes irresistible grace. I tried to please Him as a lost man, and understood salvation for years, but pride kept me from accepting Him. So there goes total inability.
Your characterization of your conversion is not at odds with Calvinist doctrine. According to most every form of non-Arminian Protestantism, every sinner resists the call of the gospel until God quickens the chosen, making them willing.
It's referred to as the outward call of the gospel, which every man resists. Then. There is an effectual call which is irresistible. That man is totally passive in this effectual call.
I'm not suggesting you become a Calvinist, because it is erroneous. But, if your own conversion is all that would keep you from embracing Calvinism, you have little grounds to do so