Are we all moving at $c$? Is it true to say that all matter in the universe is travelling with velocity c through spacetime, but that for baryonic matter most of that velocity is through the time dimensions rather than the XYZ axes?
 A: 
Is it true to say that all matter in the universe is travelling with velocity c through spacetime?

Actually, no. Because there is no motion through spacetime. Spacetime models motion through space over time, but because it includes the time dimension, it's totally static. And whilst this  model works very well, we live in a world of space and motion, not a "block universe". To appreciate this, imagine I throw a red ball across the room, and you film it with an old-style cine camera. Then you develop the film and cut it up into individual frames, and stack them into a block. There's a red streak in the block, that's like the ball's worldline. But the ball isn't moving up the block. The map is not the territory.     

for baryonic matter most of that velocity is through the time dimensions rather than the XYZ axes?

No. Time is a dimension of measure, not a dimension that offers freedom of motion. I can hop forward a metre, but you can't hop forwards a second. There's no motion through time, just as there's no motion through spacetime. A good way to "get" this is to think of the science-fiction device called a stasis box. No motion of any kind occurs in the stasis box. Light doesn't move, electrochemical signals don't move, nothing moves at all. So when I shut you inside the stasis box for five years then open the door, you think I opened it immediately. You "travelled to the future" by not moving at all whilst everything else did. However having said all that, the wave nature of matter is not in doubt, and such waves are associated with motion at c. Hence the Dirac equation:
$$\left(\beta mc^2 + c(\alpha_1 p_1 + \alpha_2 p_2 + \alpha_3 p_3)\right) \psi (x,t) = i \hbar \frac{\partial\psi(x,t) }{\partial t}$$ 
It's a relativistic wave equation which describes spin-½ particles such as  electrons and quarks. So it is reasonable to say that all matter in the universe is associated with motion at c.   
