Blotchy wood is going to very difficult to stain evenly. There's no secret recipe for a stain that won't blotch. Minwax sells a blotch controller, which is water based and works decent but not great. The wood you've got accepts stain unevenly, which is the blotching. The blotch controller works on the idea that if you saturate the wood with the blotch controller and let it sit for a minute (per directions), then the Minwax stuff will prevent the areas that accept too much stain from doing so. Therefore, the stain should even out. Unfortunately, it doesn't work that well. Other options are to go to Home Depot or Lowes and buy some of the Dewaxed Shellac (I think it's Minwax also, but be sure it's dewaxed) and dilute it by half with Denatured Alcohol and apply that to the stock and let it dry (dries fast). It will, to some degree, be absorbed more by the blotchy parts of the wood, which leads to more even acceptance by the wood of your stain. That does work, but it limits the amount of the stain that the wood will take. So use the diluted shellac, sand lightly to 320 grit, and apply your stain and if you want it darker, then use a wipe-on stain/varnish. That'll not penetrate the wood. Once that dries, go over it with a polyurethane varnish or with shellac (2 coats or more). Yes, it's a pain, but blotchy wood is hard to whip into shape.
One other approach that works sometimes, and that I've used on Cherry but not on birch, is to sand it really really fine (to 600 grit or more). That also limits the absorption of stain. It might work. It might not work.