Could you spend a golden dragon in one of the Free Cities, gold being gold, or would you have to convert it to a local currency somehow?

Depends on whether the merchant in question does a lot of business with Westeros (and thus needs dragons to buy stuff in Westeros) or not, in which case they’re going to want local currency.

The thing about gold being gold is that coins have different sizes, weights, and purities, so a dragon is going to be worth a certain number of Volantene honors and a different number of Braavosi square-coins and a certain number of Lyseni ovals, and so forth. 

Keeping track of the different types of coins and their value is often a pain in the ass, which is why merchants might either make use of a moneylender themselves or tell you to change your money first and then pay them.