I don't know what some folks call stabilizing, but the process is a bit more than complex than applying a two-pat resin. Very few home stabilizers do more than coat the wood with some sort of resin or acrylic. What method are you using?
As for the dyes, get a dye made for wood. Ritt is a water based dye...you don't want that. Use a good grade alcohol based wood dye from Woodcraft or other woodworking source. Dye does not fill the pores like paint, it stains the fibers. Also, use the right type of alcohol to mix it. Properly done, the color can penetrate right through the wood.
The best solution for a perfect job on your piece of wood, in the color desired , is to send it to WSSI and have them dye it and properly stabilize it. This is what most of the professional makers do.
Make sure it is fully dry first....below 10% moisture content ( 7% is even better).
Realize that, even with stabilized wood, there will be the ability to contain food residue and bacteria. The stabilizing will help prevent warp by stopping the ability of the fibers to absorb moisture, but the pores and channels will mainly still be there.Capillary action will still allow things to wick up into the wood. Burl woods are the worst offenders for this, as they are full of small ( and not so small) voids by nature. Stabilizing will not fill voids.That needs to be done in the finishing steps.
For food use, the surface should be fully sealed after shaping and before final sanding. There are several ways to do this. You can apply a CA finish , which takes a lot of time, but fully seals the entire surface. Bruce Bump's tutorial on CA finishes is great.
Alternately, you can apply a sealing oil, like walnut oil ( which is food grade), allow it to dry partially, sand it into the surface, let dry, and repeat until all pores are filled.This takes time,too. Other food grade finishes can be used as a sanding sealer,too. Waterlox is one of these. Whatever you use, apply it, let it set for a while, sand it in ,wipe it off, let it dry, sand any build up off the surface, dry....repeat. This will slowly seal every pore and scratch, leaving a smooth and solid surface. The final sanding should be to at least 1000 grit,2500 is even better, followed by a buff with a soft cloth by hand. Such a finish can be easily restored with a few strokes of the fine grit paper if it gets too dull.
As with all fine quality wood knife handles, it should be rinsed when used and wiped off. Then hand washed as soon as practical and dried well. This will maintain the handle long into the future. Soaking in the sink and/or putting in dishwashers will ruin almost any wooden handle.