  In the presence of neutrino Non-Standard Interactions (NSI) with matter, the
derivation of neutrino parameters from oscillation data must be reconsidered.
In particular, along with the standard solution to neutrino oscillation,
another solution known as "LMA--Dark" is compatible with global oscillation
data and requires both $\theta_{12}>\pi/4$ and a certain flavor pattern of NSI
with an effective coupling comparable to $G_F$. Contrary to conventional
expectations, there is a class of models based on a new $U(1)_X$ gauge symmetry
with a gauge boson of mass of few MeV to few 10 MeV that can viably give rise
to such large NSI. These models can in principle be tested by Coherent Elastic
$\nu$-Nucleus Scattering (CE$\nu$NS) experiments such as COHERENT and the
upcoming reactor neutrino experiment, CONUS. We analyze how the recent results
from the COHERENT experiment constrain these models and forecast the discovery
potential with future measurements from COHERENT and CONUS. We also derive the
constraints from COHERENT on lepton flavor violating NSI.
