You do, but it's too small to really notice
First, it's not correct to say that we don't feel Earth's rotation because it's rotating at a constant speed.
Think about driving a car, or riding in an airplane. Whether you're cruising down the road at 90 kph, or soaring through the air at 900 kph, you don't really "feel the speed".
However, When you take a sharp turn, or take off from the runway, you definitely feel something. That's the acceleration. It doesn't matter if your speedometer stays steady - if you take a sudden 90 degree turn, you're going to feel it.
More relaxed turns, such as going through a roundabout, or when the airplane circles the airport before landing, are much less likely to spill your drink.
Even if Earth is spinning at a constant speed, the spin is a change in direction, which requires acceleration.
Acceleration is quite noticeable, depending on its magnitude. Even just sitting down, you can feel the pull of Earth's 9.8 m/s² gravity - your body's "weight", as it were.
So how large is the acceleration keeping Earth in orbit? About 0.0059 m/s².
What about the acceleration of Earth's rotation? A ever so slightly larger 0.0339 m/s².
Small wonder that it seems like you can't feel these forces!