Especially when some monsters resist mundane weapons.
I agree. I would aim at diminishing the number of magic items, but making them more powerful.
Now I'm not 100% sure what your question truly is: do you want to know how to keep the game playable without magic items, or how to give them magic items. Maybe clearing that up would help us help you.
Personally, I would do one of two things: either (a) use Weapons of Legacy, i.e. magic items that become more powerful as you gain level, although you have to do certain rituals at some point so you become more in sync with the item (plot hooks); or (b) give them the bonuses as is, to their person rather than as magic items: either make a skill tree a la video game RPG, based on acceptable gear at each level, or just straight out give them extra bonuses when they level (e.g. at level 3, you would get +1 armor, so you get a magical +1 to AC).