Confusion with Noether's theorem: Time symmetric system with velocity-dependent terms?

Noether’s theorem says that any system that is time-translation-symmetric displays energy conservation, and vice-versa. However, I’m not sure if this is the case.

Suppose we have a two particles (of the same mass $m$) connected by a spring (of spring constant $k$) with a damping coefficent $b$:
begin{align*}
F_{A} = mddot{x}_{A} &= -k(x_{A} – x_{B}) – bdot{x}_{A},
F_{B} = mddot{x}_{B} &= -k(x_{B} – x_{A}) – bdot{x}_{B}.
end{align*}
For reference, I’ll write the solution to this below.

We’ll write $vec{x}(t) = begin{bmatrix} x_{A}(t) x_{B}(t) end{bmatrix}$ along with $gamma = dfrac{b}{2m}$ and $omega = sqrt{dfrac{2k}{m}}$ for succinctness.
The general solution to the differential equations is a linear combination of the following solutions.
First, we have the part where the two masses coincide:
$$ vec{x}(t) = begin{bmatrix} A A end{bmatrix} + begin{bmatrix} B B end{bmatrix} e^{-2gamma t} $$
where $A, B$ are constants.
Second, we have the oscillation part:
begin{align*}
&text{Overdamped } (gamma > omega): && vec{x}(t) = e^{-gamma t}left( begin{bmatrix} C -C end{bmatrix}e^{-tsqrt{gamma^{2}-omega^{2}}} + begin{bmatrix} D -D end{bmatrix}e^{tsqrt{gamma^{2}-omega^{2}}} right) [1.5ex]
&text{Critical damping } (gamma = omega): && vec{x}(t) = begin{bmatrix} C -C end{bmatrix} e^{-gamma t} + begin{bmatrix} D -D end{bmatrix} t e^{-gamma t} [1.5ex]
&text{Underdamping } (gamma < omega): && vec{x}(t) = e^{-gamma t}left( begin{bmatrix} C -C end{bmatrix}cos(tsqrt{omega^{2} – gamma^{2}}) + begin{bmatrix} D -D end{bmatrix}sin(tsqrt{omega^{2} – gamma^{2}}) right)
end{align*}
where $C, D$ are constants.

Clearly none of the forces are explicit functions of time, so the system is time-translation-symmetric. However, the energy $E = frac{1}{2}mdot{x}_{A}^{2} + frac{1}{2}mdot{x}_{B}^{2} + frac{1}{2}k(x_{A}-x_{B})^{2}$ is clearly not conserved.

According to this and this, I can convert my system to a Lagrangian system with
begin{align*}
L &= frac{1}{2}m(dot{x}_{A}^{2} + dot{x}_{B}^{2}) – frac{1}{2}k(x_{A} – x_{B})^{2}, qquad Q_{k} = -bdot{x}_{k} qquad (k=A, B),
end{align*}
and equations of motion
begin{align*}
frac{d}{dt}left( frac{partial L}{partial dot{x}_{k}} right) – frac{partial L}{partial x_{k}} = Q_{k} qquad (k=A, B).
end{align*}
This is again apparently a time-translation-symmetric system, but the usual definition of energy is not conserved.

How is this compatible with Noether’s theorem?