In a static spacetime, there is (by definition) a timelike Killing vector field $\xi^\mu$, which implies that geodesics with four-velocity $u^\nu$ have a conserved quantity $\epsilon = -g_{\mu\nu}\xi^\mu u^\nu$. For example, in Schwarzschild spacetime, this is
$$\epsilon = \left(1-\frac{2M}{r}\right)\frac{\mathrm{d}t}{\mathrm{d}\lambda}\text{,}$$
where $\lambda$ is an arbitrary affine parameter for the geodesic. This is the correct generalization of conservation of orbital energy in the general-relativistic context.
For a massive particle in Schwarzschild space, for which we can take $\lambda$ to be the proper time $\tau$, this leads to a direct analogue of Newtonian orbital energy corrected by an extra $r^{-3}$ term, up to the understanding that the Schwarzschild $r$ and $\tau$ have somewhat different meaning than they do in Newtonian theory:
$$\text{const} = \frac{1}{2}\left(\frac{\mathrm{d}r}{\mathrm{d}\tau}\right)^2 + \frac{1}{2}\frac{l^2}{r^2} - \frac{GM}{r} - \frac{GMl^2}{c^2r^3}\text{,}$$
where $l = r^2(\sin^2\theta)(\mathrm{d}\phi/\mathrm{d}\tau)$ is the specific angular momentum, another conserved quantity for Schwarzschild spacetime. The first two terms would constitute the Newtonian kinetic energy per mass, decomposed into radial and angular components.
For a massless particle, this is a little different:
$$\epsilon^2 = \left(\frac{\mathrm{d}r}{\mathrm{d}\lambda}\right)^2 + \frac{l^2}{r^2}\left(1-\frac{2M}{r}\right)\text{.}$$
In GR there is no gravitational energy so the photon did not trade "light energy" with potential energy.
But in static spacetimes, we can always define a conserved total orbital energy for the geodesics.
I just read the derivation of your formulas in Carrolls GR book. You get conserved quantities but not really an exact energy term. You get something like energy per unit mass. To turn this into energy seems critical since photon mass is zero (and we look at photons in this example). Mathematically we have a conserved quantity but I don't see how we determine that this is really the energy and not something else.
A conserved quantity generated by time-translation invariance is how conservation of energy works in modern physics; that's the moral of Noether's theorem and the geometrical meaning of having a timelike Killing vector field.
Having it specific energy (i.e., per-mass) is exactly how gravitational potential works even in Newtonian theory, and even for a photon it can be interpreted as relative to energy at infinity, or adapted to a scale set anywhere along the orbit, for that matter. Operationally, it is exactly the specific energy measured by a family of static observers (comoving with the Killing vector field), since an inner product with a four-velocities gives the relatively Lorentz factor, or equivalently, the time-component of one in a local inertial frame comoving with the other.
