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As a softwood, pine does not absorb wood stain evenly. Simply applying a stain with no preparation may result in a "blotchy" appearance with contrasting dark and light areas. Applying the wood conditioner equalizes the absorbency of the pine so that a more even finish is obtained. Follow the directions on the can regarding proper application of this product.
Be aware that if you wish to apply a second coat of stain to darken the finish, you should apply the wood conditioner AGAIN prior to restaining (especially if you wait overnight before applying the second coat) or the stain may not absorb evenly.
Wood harvested at certain times of the year is more likely to exude sap from the knots, even when kiln-dried. It isn't considered a defect, though it can be a pain!
I would suggest just removing the sap... for now... to get rid of the stickiness. Once you go through a season or two without more sap appearing, you can refinish the post. Just sand the affected side and apply a coat of polyurethane to it with a brush or cloth. There are rub-on polyurethane finishes that are easy to use and work well! Make sure you don't apply any product that contains wax or oil to the posts or the new poly may not stick!!
There are a number of sap-removal products you can try. Denatured alcohol is the least damaging to the polyurethane. If that doesn't do the trick, try automobile finish sap remover. Other products that will remove stubborn sap are kerosene (lighter fluid) and gasoline, though I wouldn't recommend using gasoline inside your home.
Stains of that era were most likely hand-mixed and colors were not as consistent as modern manufactured stains. So forget finding a color match by name!
I suggest getting a piece of similar pine and a few cans of Minwax or other brand of wood stain that closely match the original... and test them on the scrap pine. If necessary, intermix them to obtain an intermediate color. I have done this successfully in the past and, since you are going to do the entire staircase matching is less of a problem. The resulting color should look great!
Just don't mix stain brands OR mix oil-based with water-based stains. Who knows what brew you might come up with!