For hand calculation around 10-20% of accuracy isn't a problem, especially for CMOS nodes older than 130nm, so if You want to working in strong inversion, a square law s+hould be ok and quite accurate if Vds/L < 0.8MV/m for nmos and 1.95MV/m for pmos (values of critical field E_C).
If You really want to obtain good accuracy You should use one of the physical models, i.e.
EKV or
ACM which are quite easy to use ;-)
In saturation region drain current is described by formula:
\[I_d = i_f I_{spec}\]
where I_spec is a specific current for given technology and mosfet dimensions and in EKV is given as (in ACM factor 0.5 exists instead of 2):
\[I_{spec}=2nV_t^2 K \frac{W}{L}\]
and i_f is a normalized drain current which is bounded with normalized charge in channel by following relation:
\[i_f=q_f^2+q_f\],
while q_f is given by Lambert W function:
\[q_f=\frac{1}{2} W \left ( 2 e^{2 + \frac{V_{GS} - V_{Th}}{n V_t}} \right )\]
The K is current gain factor K=µCox and every effects with higher vertical/longitudinal field and series resistance is modeled as a factor in K:
\[K_{eff}=\frac{K_0}{[1+(V_{GS}-V_{Th})(\theta + K_0W/L(R_S+R_D))][1+(\frac{V_{DS}}{L E_C})^\beta]^{1/\beta}}\]
where \[\theta\] is constant in order of 1e-7/tox, E_C is above, \beta is equal 2 for nmos and 1 for pmos fets, Rs=Rd for rectangular transistors and is equal to contact resistances (parallel connection of vias at drain and source).
The DIBL You can modeled by varying the threshold voltage:
\[V_{Th}=V_{Th_0}-V_{DS}e^{-\lambda/L}\]
where \lambda is a "characteristic length" defined in Tsividis book about modelling (factor of many technological constants)
Using above equation You are able to solve it for given properties by hand (or by Mathemitaca tool ;-) ) or by some numerical calculations in C, python or other language.
If You really want to study about mosfet modelling, check papers about EKV, ACM models and also about BSIM6 which are develope from EKV.
Some links:
https://www.sciencedirect.com/science/article/pii/S0038110111003492
https://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6343331&tag=1
https://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6632892
https://www.sciencedirect.com/science/article/pii/S0026269214001323