PB Blaster smells bad, but loosens rust extremely well. Squirt a little on the screws and let it sit for a day. Maybe do a refresh squirt half way inbetween. The screws should come off.
I don't know how bad your action rust is, but only Browning can exactly match the original finish. If you have significant pitting, it may never match exactly, but that won't likely affect the functionality of the rifle. The same PB Blaster will loosen all the rust you place it on, so you can often wipe a lot of it off, then use a toothbrush to get more of it out.
The only way I know to chemically neutralize rust without damaging bluing is to get it degreased with solvents (a soak for a day in odorless mineral spirits that are then allowed to dry - it may also require flushing with Bore Scrubber or something similar) then boiling in distilled (to avoid water spots) water, which will convert rust from red to black form, most of which will rub off, but which does not cause further pitting. If the rust is too thick or deep or has oil in it too deeply, this will not work well as the water can't get into intimate contact with the deep portion of the rut. If you are going to try this, I suggest you do it after removing the scope, but before getting PB on the action. After the boiling (about 15 minutes will do all it can), then use the PB Blaster soak and toothbrush approach. It will remove the loose blackened rust and will reveal if unconverted rust is underneath.
Damprid has not only calcium chloride, but sodium and potassium chloride as well (it's hard to purify calcium chloride completely), all of which cause rust if they come into contact with steel. It is in pellet and flake form. It swells and hardens as it takes on water, but if you have liquid in the pouch, the stuff is used up and is overdue to be replaced. This is described in the company's product FAQ, 4th item down,
here.
