Hi,
Just an update; the cactus is fine! Four days ago I cleaned the bottom best I could with a clean, unused paintbrush. I found some fairly concentrated solution of colloidal silver in the house, so I thought what the hell (don't worry I did my research). I sprayed and rubbed it onto the rots to stop bacterial growth and they dried out completely by the next morning. I repotted it today and I will not water till a week later.
I should mention though the effects of silver uptake through roots from the soil are unknown, so you know, just a warning before anybody gets any ideas.
I think I will give an update in the future when and if there is an irrefutable sign of improvement, in case others with the same problem see this thread.
Thanks again for the helpful advice!