Increasing surface area has no effect on buoyancy. Changing volume does have an effect. When I was young and skinny, I could sink to the bottom if I just let most of the air out of my lungs. But I still couldn't float properly even with full lungs until I learned to push the air in my lungs closer to my center of mass. If you don't have enough lung capacity to fill your lungs with air and thereby reduce your mass-to-volume ratio to less than that of water, you will sink. If your buoyancy is slightly positive and your center of mass is below your center of buoyancy (tends to be somewhere in the lung area), then your feet will sink and the top of your head will stay just above the water surface.
Most people tense their stomach muscles when they're not relaxed. If you relax your stomach muscles, your diaphragm can move downward (that is, toward your navel) and thereby move your center of buoyancy the same direction. Alternatively, if you stretch your arms above your head (that is, away from your navel), you're moving your center of mass upward toward your center of buoyancy. Full lungs, arms above your head, and relaxing your belly muscles should make you float just fine.