The bronze axe isn't going to help you much, without a forge to repair it with... and that requires 6 copper. Just go mine up what you need, either by killing Eikthyr to get the antlers with which to make a pickaxe, or by aggravating trolls, and raid a burial chamber or 3 to get the 5 or 10 surtling cores you'll need for the charcoal kiln and smelter.

You've literally spent more time begging for a handout than it would have taken to simply go and get what you need; it doesn't sound like you're "rushing" anything.