Sand blasting
I use a sandblaster to recover old and rusted rifles and have had good success. This is an old dodge for restoring antique (eek!) guns. The result is that rust damage and pits are not very noticeable.
Sand blasting is great for removing rust and crud. It is however, very abrasive (go figure) and will remove metal quite quickly. Don't blast any surfaces that have a fit with each other. Blasting also tends to be a bit harsh and leave a rough finish if the sand is of a course grade. Very fine sand works best, or even better, bead blast it. The tiny glass beads are more gentle. Just be careful of sand in the working bits. Remove any mechanism, sear etc. Or use duct tape to mask off and protect things.
The surface should be buffed afterwards with steel wool and then degreased before applying the cold blue. The surface, which will be clean bare steel will just 'suck up' the blue. I have had very good results. The finish is dark and has a nice flat finish, not unlike worn Parkerising. Follow the directions on the bottle to the letter.
Take your pick as to which cold blue to use, there are dozens to choose from. I find that they all are much the same. Perhaps somebody could recommend one brand to you.
I recommend that you do some trials on scrap pieces of metal before you blast the barrel and action. The trick is in the surface preparation. Blasted, sanded or polished, all produce different finishes.
Oil it well with a good gun oil to protect it.