I've had a few archer characters and haven't seen anything like this even after completing the MQ and a significant portion of others.
Sounds like a corrupt save file or misbehaving mods.
Both characters are using the same mods... Corrupt save is possible, but I dunno... Worth mentioning that I'm using Faendal as a follower, who is also using a bow.
Here's my mod list, if you see any issues feel free to point them out...
Spoiler
Update.esm
Shadow Striping Fix.esp
Unofficial Skyrim Patch.esp
SoS - The Dungeons.esp
Realistic Lighting.esp
Realistic Lighting Patcher.esp
enhancedlightingforrealisticlighting.esp
Natural Skyrim Rain.esp
arrowsmith.esp
Fixed Followers.esp
KillThemGenerals.esp
new guards.esp
Mart's Savegame DeBloatifier.esp
minemapmarkers.esp
Dragonbone Weapons.esp
Magicka Sabers.esp
Smithing Perks Overhaul - Balanced.esp
Weapons and Armor fixes.esp
Weapons and Armor fixes - Hacks.esp
recipeingots.esp
Complete Crafting Overhaul.esp
PISE.esp
DragonboneWeightReduction(x3).esp
lightertools.esp
lightweight potions and poisons.esp
waterbreathing breathless emerge.esp
Better Dynamic Snow.esp
betterquestobjectives.esp
Lush Grass.esp
torches for realistic lighting.esp
Detailed Cities.esp
dD - Realistic Ragdoll Force - Realistic.esp
Combat Perks Overhaul.esp
betterwerewolf.esp
better vampires.esp
MidasSkyrim.esp
Divine Intervention.esp
Trader Taklas.esp
Balanced_Alteraton.esp
Balanced_Conjuration.esp
Balanced_Destruction.esp
Balanced_Illusion.esp
Balanced_Restoration.esp
Balanced_Magic_Adept.esp
portal2mod.esp
Shadow Striping Fix.esp, Divine Intervention.esp, and Trader Taklas.esp are created by me, so you can ignore those.