  Two quantum-corrected black hole models have recently been proposed within
the Hamiltonian constraints approach to quantum gravity, maintaining general
covariance \cite{Zhang:2024khj}. We have studied the quasinormal spectra of
these black holes using three methods: the higher-order WKB approach with
Pad\'e approximants, time-domain integration, and the Frobenius method. The
Frobenius method, in particular, allows us to determine precise values of the
frequencies, including the overtones. The two models differ in their choice of
quantum parameter $\xi$, and we can distinguish them by their quasinormal
spectra. In the first model, increasing the quantum parameter results in higher
real oscillation frequencies and damping rates of the fundamental mode. In
contrast, the second model shows a decrease in the oscillation frequency of the
least-damped mode when the quantum parameter is introduced. We have shown that,
while the fundamental mode changes relatively gradually with the quantum
parameter, the first few overtones deviate from their Schwarzschild limits at
an increasing rate. This results in a qualitatively new behavior: the real
parts of the frequencies of the first and higher overtones tend to zero as the
quantum parameter increases. In addition to the branch of modes that are
perturbative in the quantum parameter, we observe some non-perturbative modes
at moderate values of the quantum parameter.
  Additionally, we have calculated the radii of the shadows cast by these black
holes and discussed possible constraints based on observations of Sgt $A^*$.
