Only when TOS Enterprise is actively maneuvering or turning do I recall seeing her point in a different direction; I've never seen the Enterprise warping through space facing sideways.
Once she's settled on course, she's always going forwards or backwards. And she goes backwards only in an emergency.
I agree for most "normal circumstances", when she's traveling somewhere she's going to go forward. My guess is that's probably the most efficient for her design. But she's quite capable of pointing in a different direction other than her flight vector at warp or at impulse if there was a reason to. Usually that means pointing her forward phasers or some sensor gear towards an object.
Alot of the shots of the TOS Enterprise (original FX) shows her flying "not quite" straight forward but at a slight tilt up. Her opening intro sequence where she flies towards the viewer shows a curve in her flight path although the ship never turns or pitches in direction to change her flight path.
When Enterprise is traveling to intercept V'ger in the TMP: Director's Cut version where Kirk, McCoy and Spock are discussing something you can see in the back viewport that the ship is traveling slightly "sideways" as the vanishing point for the passing "stars" is not directly behind the ship relative to the warp nacelle.
Not Normal circumstances:
In "The Paradise Syndrome" she's flying sideways to the asteroid when she attempts to deflect it. Then she moves directly in front of the asteroid flying backwards intending to split it.
In "The Corbomite Maneuver" she's spiraling, then moving forward and then reverse and then reverse at warp trying to distance herself from the cube. Later on, Enterprise breaks free from the tractor beam by shearing away at a right angle course.
In "The Doomsday Machine" one of the combatants is either flying forward or backwards relative to the flight path of the Doomsday Machine.
When Enterprise intercepts V'ger in "The Motion Picture" she approaches flies over at 500m paralleling V'ger all while V'ger is traveling at warp speed and then settles in front (or behind) V'ger and reverses direction to face V'ger. You can take it either Enterprise flew in reverse to fly over V'ger or flew forward and reversed direction once she got in front of V'ger.
Initially Normal circumstance:
When Reliant intercepts Enterprise in the first battle in TWOK, Enterprise is traveling forward and Reliant is approaching in reverse. Forward motion is maintained and Reliant is essentially flying sideways relative to Enterprise.
There are more examples, but those are the ones I can think of right now