I have used Minwax stains for many years and have experienced that same issue. There isn't anything wrong with the product; I have found its the application technique. I use to "flood" the work and then wipe off after a few minutes. That flooding is what fills the pores and later bleeds back out. I switched to a lighter application using a rag or a dryish foam brush. I then wipe off as soon as I finish applying to an area. This typically takes more than one stain application but it has virtually eliminated my bleed out problem. I have noticed that sapwood does this worse than heart. I sometime get bleed out even with this lighter touch if the piece has any sapwood in it. Once you start getting bleed out you're stuck until it stops. I have gotten impatient and top coated a piece I thought was done bleeding only to find it still bled out under the finish.

I have also had trouble with my top coat adhering to a piece where I have left much of a stain film on the surface. I have this problem regardless of how long I let the stain set. Happens with poly and lacquer. I typically stain red oak to a brownish mahogany to a reddish walnut and its hard to achieve that color without leaving a surface film of stain. I have changed my technique (based on input received on this forum). I now try and get my surface color with a water based dye/stain, and then add depth with a very light coat of minwax oil stain. I'm still adjusting this technique but I like the result a lot and have almost none of the headaches I use to have with Minwax stains.