OK, I went on Minwax's Q&A site. Apparently, I did not rub away the extra stain hard enough, which left the extra to become tacky. I used mineral spirits to wipe away the stain. Back to square one, I guess.

Did you sand off ALL the previous finish or just rought it with sandpaper? MinWax oil based stain will not stain over a finish, it has to be applied to bare wood.

Also, stain has a shelf life, if it is too old, it will not dry properly.

Try getting the stain from a Sherwin Williams paint store, their stock is usually fresher.

You know, I can't bet my life that I sanded whatever finish was on the railing. I thought I did, based on my previous work on a table and chairs, but apparently not so maybe. I just bought the stain from HD, so not sure how old it is. I'm still working on removing it with the mineral spirits. Just when I thought it's done, it still feels tacky afterwards. :(

You clearly failed to remove all of the existing finish. Were you applying stain to bare poplar, it would have dried to the touch in 2 hours.
I would scrap the liquid stain altogether and use a gel stain. No pre-conditioner is needed.
Casey

Thanks everyone. Chalk this up to a learning experience.