There's no reason why the 2 rates should move in the same direction - these are 2 separate markets.
To see this, think of a different asset class, say residential housing: housing prices may be falling, the cost of fire insurance for a house may be going up.
Keep in mind that the interest rate is the price of credit, not the price of money; the exchange rate is the price of money. A CDS - being the price of insuring principal and interest payments for a security - doesn't depend on demand or supply of credit, it depends on the uncertainty surrounding the payments - whether they will be made as promised, first, and, second, whether, when made, they will be worth the same (in terms of foreign exchange) as today. You can think of gold as a forex proxy, if that helps. If the second concern is uppermost in your mind, look into Treasury inflation-protected securities (TIPS).