I think you are right!You have pretty old hardware there, I don't think they have put ball floats in toilets for decades. Doesn't mean they don't work though. Seems your problem is only the flush area not the fill. It is running (as you have figure out) because water is leaking out of the tank and into the toilet.
Don't usually use sealant, the gasket is held down by the plastic nut that is threaded around the fill tube. Maybe it just needs to be screwed down a bit tighter?
From the look of it, you might have one of these, I've not seen one before. I think it would have a nut on the underside of the tank, not shown here:
https://www.acehardware.com/departments/plumbing/toilets/toilet-repair-parts/4591533
Can you see a plastic nut between the tank and the bowl? If so, can you tighten it while holding the part inside the toilet so it doesn't turn and lose the proper angle to the flush lever?
I don't think that will work with what SGC has in her tank.
I would replace all of it though and go dual flush
. I was so ticked. Sure enough, when the gal at work saw her, asked why she was there. Had to help her through a lot of being sick cleaning her up before calling and telling those people to come take her home. When I found out...I had some talking to do. She is so sick. She is now in stage 5 kidney failure. I called Drs. office ..Nephrologist..talked with his assistant. Told her I had heard three different opinions going on with Lisa..most concerning. Not eating. Can't keep fluids in her..sleeping sleeping, sleeping. This assistant called the Dr. He said to get her to the hospital. Thank heavens. They say her kidney function is still ok to be home. They did find a UTI. She has an appointment on Monday with the Nephrologist. She dropped so much in less than a week..to a 6. ..I'm sure hoping they put that port in. She isn't going to make it 3 more months waiting for that fishula to be ready! Went to visit her today. She had , I would guess, 5 or 6 blankets on. Asked, where is your kitty blanket? The one SCG made for her. She says in a sad voice, I don't know, I haven't seen it since I moved here. They moved her a few months back. I found someone, Lisa actually walked with us. She knew just the very cabinets for them to check. Second cabinet opens, she sees it! She grabs it, turns, and walks back to her room. 
She now has around 6 blankets on. She's always cold.. kidney failure does that. Oh man, it's good to show up at her place every now and again to keep things straight..and..phone calls to the head nurse!
