Wilderness Systems makes a boat called the Shaman ($750) that probably fits the bill. It’s a fairly short (12 feet, three inches) touring-style kayak with an enclosed cockpit (meaning you can fit a spray skirt around it) and rear storage area with a sealed hatch. And it has Wilderness System’s multi-adjustable Phase 3 seating system, which is nice. Wilderness Systems doesn’t give the Shaman a whitewater rating, but Class II should be well within its capabilities.
Dagger also makes a kayak that’s specifically designed to meet your task description. It’s called the Crossover, appropriately enough, and is a 12-foot, six-inch boat that can be configured skeg down for calm-water touring, and skeg up for rougher stuff. Dagger says it can take up to Class III. A rear hatch provides access to enough storage space for an efficiently packed three-four days’ worth of gear. Price is reasonable at $889. You’re not far out of average in terms of height and volume yourself, so you ought to fit.
Perception’s new Antigua ($639 with skeg) is similar to the Crossover-just a tad shorter and designed for touring on calm water or rivers up to Class II. Even has a cup holder, so you can keep your morning coffee handy!
Any of those three boats should do you right. If you can, give each a try to see what fits you best and feels best in the water.