Well, that’s not much more of an anachronistic mess than the actual 19th century.
The best way to level things out would be to remember that all not weapons are created equal. I don’t mean some weapons are better than others, this is an easy mistake, I mean weapons have different uses.
Put it this way, you wouldn’t try to drive nails with a screw driver or a pry bar, and you wouldn’t try to remove a screw with a claw hammer.
Weapons are tools. You might misuse them if you’re desperate, but ultimately, your characters need to use the most appropriate weapon for the current situation. Try to use a weapon in the wrong situation and your characters will die.
The thing about guns is, they tend to be more versatile than other weapons, and that’s not something you can get around. They’re also more forgiving of being used in the wrong role. Someone who has a rifle can still kill at a foot away, the same is not true of bows or crossbows. Try to ignore that and your writing will suffer.
Handguns are most effective in very tight spaces, where long range accuracy isn’t an issue. They also work in situations where you need to conceal a weapon. Try to fight someone more than fifty feet away, and your character will die. (You can triple that number for modern handguns.)
Swords (in that timeframe) are most effective in mass melee. They’re also (slightly) stealthier than the handgun at those ranges, and can offer a significant advantage if their foes don’t realize they’re there. Try to go up against any aware foe with a firearms; your character’s going to become chunky salsa, and they should know that.
Rifles are best for long range precision shots. Granted, 19th century firearms left something to be desired when it came to accuracy, but these were still weapons you could use to out range combatants. These start to fall apart at close ranges, and trying to take a rifle into a cramped house or tunnel system can be fatal.
Bows and crossbows are still the top choice for silent killing at range. They don’t work at the long range of the rifle, they can’t be used in close quarters, the way a handgun can, bows are very hard to operate under fire, and crossbows take awhile to reset, but that’s not the point. If no one knows your character is there, it’s a good choice, if anyone can scream in agony as they’re dying, your character better have a fallback weapon.
Also, lever and bolt-action rifles date back to the 19th century, so those aren’t automatically out. As do breach loading and lever action shotguns.
Lever-action repeaters trade some of the accuracy and range of a rifle for maneuverability and follow up. Bolt-action rifles are similar, though they keep a lot of the long range accuracy. Both are cumbersome to reload, though the same is true of most 19th century firearms. Again, neither will help you if you’re getting run through from behind with a sword or picked off with a bow, or (with repeaters), if someone is taking shots at 300 yrds, but, they will ruin someone’s day if all they have is a revolver, a sword, or a bow.
Shotguns operate at roughly the same ranges as a handgun, they trade maneuverability for stopping power. Games will give you a bad idea about how much shotguns actually scatter, but if you want something dead right now, and it’s not too far away…
If you can stomach it’s writing and “white savior” racism, Far Cry 3 is a massive playground to get the idea of the right weapon for the right situation down. The roles it places the weapons in aren’t 100% accurate, but it might help you grasp how these things interact in a modern setting.