Should I disassemble the lock before applying browning or darkening solution?
Yes, it is much better to disassemble the lock. You don't want to darken the internals or the inside of the plate. With a mainspring vise it's not hard to take apart. If you proceed without disassembling, use swabs carefully on exposed surfaces only, keep solution off the internals, and apply plenty of oil afterward. Track of the Wolf sells a mainspring vise that works well.