How long are you waiting between coats? When you apply the solution LESS IS MORE. Give it a wipe and that's it.
WHAT HAPPENS IF I GAVE IT A WIPE AND MISSED A SPOT?
Don't go over it again, multiple wipes are what creates that copper color and once that shows up it is a long process to get rust- you are better off going back to the white and starting over. If you gave a barrel flat a wipe and missed some of the metal- GET IT NEXT TIME.
The other thing is carding. Lightly wash with a terry cloth towel but don't use steel woool, etc- you be taking off the rust you are trying to build up.
So... Quick wipe, wait maybe 4-6 hours, hot water and terry cloth to take any possible loose rust, let dry, repeat.
The longer you leave the solution on (8 hours, etc) the rougher but thicker the rust. Usually after 4-6 coats the gun should be looking pretty good.
Finally, after you have done about as good as you can, adding linseed oil or beeswax will sort of blend the colors. Test on the bottom of the barrel (that will be covered by the stock) to see whether you prefer beeswax or linseed oil. and, this is done after neutralizing the barrel after the final acid. Use baking soda.