You are correct, the four vector (velocity) is defined as a four vector in four dimensional spacetime, that represents the relativistic counterpart of velocity (3D).
Physical events correspond to mathematical points in time and space, the set of all of them together forming a mathematical model of physical four-dimensional spacetime. The history of an object traces a curve in spacetime, called its world line. If the object has mass, so that its speed is less than the speed of light, the world line may be parametrized by the proper time of the object.
The four velocity is the rate of change of four position with respect to proper time along the curve.
The value of the magnitude of the four velocity (quantity obtained by applying the metric tensor g to the four velocity U) is always c2.
In SR, the path of an object moving relative to a certain reference frame is defined by four coordinate functions, where the timelike component is the time coordinate multiplied by the speed of light.
Thus, you are correct, you could say that us, who have rest mass we do experience time, and we are moving at the speed of light in the temporal dimension.
We just have to accept that the universe is built up so, and the four vector is built up so, that its magnitude is c always.
- spatial speed affects temporal speed
In your case, as you start moving faster in the spatial dimensions, you have to slow down in the temporal dimension, because the magnitude of the four vector has to be c always. This is the very reason that we talk about spacetime, and not space and time. This is the very reason that spacetime consists of causally linked space and time.
As you realize that your relative speed in the spatial dimensions affects your relative speed in the temporal dimension, you realize that space and time are not independent anymore.
Your example is about spatial speed affecting your temporal speed. But the two are causally linked the other way around too.
- temporal speed affects spatial speed
This is GR time dilation and when you realize that if you place an object into a gravitational field (stationary relative to the source of the field), then the object is relatively at rest in space, still it is moving through time at speed c. The four vector's magnitude is c always, thus, if the gravitational zone (gravitational potential) affects the temporal speed of the object, thus slows it down in the temporal dimension (relatively), then its spatial speed needs to compensate. What will the object do? It will start moving in space (relative to the source of the field). This is because the universe is built up so and the four vector is built up so, that its magnitude needs to be c always. If the temporal speed of the object reduces because of the gravitational field (relatively) then its spatial speed needs to compensate. The object will start moving along a geodesic towards the center of gravity. This is when you realize that time can affect space too (space curvature affects time and that affects the objects spatial dimension), thus we need to talk about spacetime as a causally linked phenomenon made up of space and time.