Note that both gems and furs give happiness. If you cancel the trade, you'll gain happiness from the gems, but lose it from furs. However, gems give an additional happiness in cities with forges; furs an additional happiness in cities with markets. Which resource is better for you depends on how your cities are developed.
Finally the trade gives you a diplomatic benefit. You can get up to a +2 diplomatic modifier for "We appreciate the years you have provided us with resources". I'll sometimes swap equivalent resources, for example, clams for crabs, just to get the diplomatic benefit, even though the trade makes no sense otherwise.
It's generally better to trade your excess if you can. The AI won't trade it's only gems, fur or whatever unless it's your vassal. However, if you have to, your strategy idea of trading your only resource for something else for the diplo benefits is a good one.
Also, the AI isn't too smart on its trades. In mid-game it assigns the same trade worth to Cows and Corn. However, until Supermarkets Corn give +2 Health and Cows give +1 Health (this presumes Granaries are in place, which they should be).
Trading for Strategic resources is a whole new topic. Those cost and are worth a LOT more.