I have read that GR predicts that moving, massive objects emit some of their energy as gravitational waves.
No, GR does not predict this. If an object is moving inertially (i.e., in a straight line with constant speed), then it doesn't radiate. The easy way to see this is that we can pick a frame of reference in which the object isn't moving at all.
An accelerating object may radiate. There are technical conditions that have to be satisfied if there is to be radiation, and there are some subtleties involved in defining what qualifies as a radiation field, but basically we expect masses to radiate when they're accelerating.
My question is this: if we set an observer into motion, and he emits some of his kinetic energy as GWs, can he in principle detect those GWs and feel a decelerating force?
If the observer is accelerating, then GR predicts that there will be a back-reaction force from the observer's own gravitational radiation. This is the gravitational version of a well-established effect in electromagnetism. Yes, the observer can measure the force. The Hulse-Taylor system, which you refer to, is an example.
However, this is not a decelerating effect on an object that is simply moving. That wouldn't make sense. Just as a matter of Galilean relativity, observers in different frames of reference don't even agree on whether an object is accelerating or decelerating. (Consider the case where you're driving alongside someone on the freeway, and then they step on the brakes.)