You could tell your dad you really want to go, but your mom is going to need proof that he's not using non prescribed drugs or alcohol. He would be able to do this by having a lab test his blood, every second day for a few weeks. (that might be long enough for your mom to trust that he's not using.) I actually self initiated blood/urine testing every second day for over two years to win a child custody case. There's no refuting a blood test to prove that you're clean and sober. Ask him if he would do that for you. I'll talk to him about it on your behalf, if you wish (through Medhelp). You can't blame your mom for being cautious. She needs proof that it will be safe for you to go visit your dad, and blood testing is the usual route for a dad that's serious and willing to go to any length to prove themselves. If it doesn't work out before you go to college , time will fly and you'll be able to visit soon enough, so don't get angry at your mom for caring that you never get hurt the way she was by your dad. Blessings.
The thing is, your mom has been there for you for 10 years. Her pay back for her dedication to you, her loyalty, her caring for you all this time is that you leave to go live with someone who has been the opposite. That would hurt. I certainly understand that you'd be curious about him and I think when a child is not around one parent much at all, they have a wound. And you want your wound to heal. But the wound is there because of HIM. He could have been in your life more. Wasn't. She was. You risk a lot leaving your primary caregiver. I couldn't recommend you leave your mom's care. good luck
How do you know he has changed, have you heard this from reliable other people who know him? Or only because he has said so and been really sweet in online communications?
I assume you probably know that alcoholics are very bad at actually changing and stopping drinking, and very very good at persuading people to hear what they want to hear, especially if the person is sad at the estrangement and really wants to believe that the alcoholic has changed.
Is he in AA? Has he done a 12-step program and acknowledges he is powerless over alcohol and has quit totally and has been in recovery for a while?
If not, please don't let him fool you with sweetness and promises. One of abusers and alcoholics' favorite things is the part after the abuse, where they devote all their energy to convincing the person that all is really better and the person should trust them again. When the person finally caves in against their common sense (that warns if they did it before they will do it again) it is a huge power trip for the manipulator.
That said, you should be able to visit your father, and can freely do so once you turn 18. If you feel strongly about going at 16, try to find a way to do it where someone older than you is with you, who can haul you out safely if things get bad, emotionally fraught, or dangerous. Could an aunt go along, and could you two could stay at a hotel versus his house? I really want you to see that going to visit someone with a history of alcoholism coupled with abuse, if you have no hard proof (besides what he has said and what you would like to believe) that he has stopped drinking, could actually wind up hurting you.
Don't go if there would be no way home but him, or if you wouldn't be able to get home except on a later day even if it turns out to be a bad situation (such as, if you had a non-changeable air ticket) and especially don't go if you have no protective adults nearby to turn to if the situation turns out to be wrong. I would be frightened for you going alone.
If you ask your mother and she says no, I don't think I'd be mad; she is dealing with her memories of him being out of control and hurting her. Being 18 is not that long from now; you can make your own choice then. But she might know more than you do on this one. You could try asking her if she would go with you, and the two of you stay at a hotel near his house, and her drop you off for visits with him and pick you up several hours later. It might open the discussion a little.
Good luck.