  Solving field equations exactly in $f(R,T)$ gravity is one of the difficult
task. To do so, many authors have adopted different methods such as assuming
both the metric functions, an equation of state (EoS) and a metric function
etc. However, such methods may not always lead to well-behaved solutions and
thereby rejection of the solutions may happen after complete calculations.
Indeed, very recent works on embedding class one methods suggested that the
chances of arriving at the well behaved-solution is very high thereby inspired
us to used it. In class one approach, we have to ansatz one of the metric
potentials and the other can be obtain from the Karmarkar condition. In this
paper, we are proposing new class one solution which is well-behaved in all
physical points of view. We have analyzed the nature of the solution by tuning
the $f(R,T)-$coupling parameter $\chi$ and found that the solution results into
stiffer EoS for $\chi=-1$ than $\chi=1$. This is because for lesser values of
$\chi$, velocity of sound is more, higher $M_{max}$ in $M-R$ curve and the EoS
parameter $\omega$ is larger. The solution satisfy the causality condition,
energy conditions, stable and static under radial perturbations (static
stability criterion) and in equilibrium (modified TOV-equation). The resulting
$M-R$ diagram from this solution is well fitted with observed values of few
compact stars such as PSR J1614-2230, Vela X-1, Cen X-3 and SAX J1808.4-3658.
Therefore, for different values of $\chi$, we have predicted the corresponding
radii and their respective moment of inertia from the $M-I$ curve.
