For Weaponsmithing in New World the resources you obtain through using Gathering skills mostly. You have no other option but to simultaneously level your interlinked skills.

Downgrade if you can and the idea itself does not freak you out. You will obtain the materials which you can use again to level your Weaponsmithing skill.

Check on prices for the materials on the Trading Post. You can get resources cheaper and then craft them. It is much faster, especially when you craft in bulk. Actually, buy iron ore if it’s cheap, craft it to the iron ingots and then use it in Weaponsmithing. Such a method will help you simultaneously and easier leveling.

When Weaponsmithing in New World stick to the Weaponry that requires lesser or easier to get materials. That will allow you to craft in bulk, so you can level up fast. You will unlock new Tier weaponry when your level rises, so opt for the easiest in the Tier you just unlocked to level up the skill.
When crafting you can use Azoth and hope for the unique perk to appear in your weaponry. Although it is not guaranteed with Weaponsmithing in New World and depends on the chance.

Manage your storage. You have three options nowadays, at least for the time being they are not blocked. You can store the items in a particular settlement and level there. Then there is an option to transport your materials from one settlement to another for some Gold, and you can use it not to travel every time. The third is…well, you can use Trading Post. Place a selling order for one day (if you want fast access), when the order will expire, you will get your items back in the storage where you currently are. This can help you with additional carry capacity, just place a price higher than the market one.