The cold blues will never match the original well, and you are likely to end up with a mottled finish trying to use them on the .22. If you are planning to strip and re-blue anyway, though, you won't hurt anything if you want to try one. The acids in them will tend eat into the original blue so you may be able to get some blending, but just don't count on it.
If you don't have hot bluing equipment, it won't be worth the investment to do yourself. You can save money by doing the stripping and polishing prep yourself, then taking the gun to a bluing operation and letting them run it through their tanks.
The best home blue, in my experience, is rust bluing, but you will need a tank big enough to let you submerge and boil the parts in water. If you do the bluing without taking the barrel off the action, you will likely want a tank in which you can submerge the action and start of the barrel in water displacing oil to help remove any water that gets into the gaps between the barrel and receiver (though just heating with a hair dryer can drive a lot of that out if the heat from boiling didn't do it.
There are also paint-on gun finishes and Parkerizing you can apply at home, though I prefer real blue for its looks, myself.
For the shotgun, try using Gunzilla
CLP. It removes trace rust very well if you apply it, let it sit a day, then wipe off with a cloth. Keep repeating that until it's gone. If you have a boiling tank for rust bluing, you can first degrease the gun thoroughly with a solvent soak, then boil it to convert trace red rust into black magnetite. Then go to the Gunzilla routine. That will blend colors and loosen the excess rust.