1) Wood is an extremely poor material to use as a sink - even with numerous coats of poly. If you're dead set on using this piece for a sink look at Water-Lox for a finish/sealant https://www.waterlox.com/original
2) Different types of wood absorb stain differently. Using one coat of stain on multiple types of wood will produce different shades/colors of stain on the different types of wood. To get a uniform color, pros spray on a coat of a toner or shading stain. You can get close to this by buying a couple bottles of a PreVal bottle sprayer. Total cost for sprayer and 2 bottles would be less than $20. Theyre available in paint and big box stores. Make a mix of 3 parts of finish, 1 part of thinner and 1 or 2 parts of stain (depending on whether you're using a dark or light stain). Spray on surface and let dry overnight. If different types of wood look similar, you can then apply a coat of stain. If there's still a lot of difference spray on another coat of the stain mix, let dry, stain and then finish
3) The reason you apply finish to wood is to keep out moisture. Wood used for furniture is dried or cured to get rid of moisture. So when people say "wood looks dried out", that's the way wood is supposed to be. Don't worry about it.