@marc_Katz The best bet is not use TIP120 Darlingtons which have a spec of:
Vce(sat)=2V @Ic=3A or an equivalent Rce of 2V/3A= 2/3 Ω .
Instead use a logic level Nch MOSFET that runs off Vgs=5V ,which is at least 3x the gate threshold voltage , and has >= 8x the current rating and an RdsOn of << 100mOhm with a reverse body diode for clamping the Back EMF.
This gives many SMT solutions, but if you prefer leaded parts consider https://www.digikey.com/product-detail/en/infineon-technologies/IRL2703PBF/IRL2703PBF-ND/811700 ($1) which is a 40 mΩ, 24A TO-220 part.
- At 3A it will dissipate < 0.25W
- and use twisted pairs to reduce EMI when switching off the solenoids with good decoupling caps at 12V source.
As warned correctly by @Janka this is more of a mechanical problem with force, mass and mobility with tradeoffs for acceleration, force and stroke length depending on mass and momentum of inertia and final kinetic energy. You might have to re-think the problem in these terms then define the power and energy needed to do the job as well as humans with pull tendons and muscles. So a gear motor might be a compromise with current sensing and/or rotation sensing for end stop with more complex control functions, or a bigger brute force solenoid that requires holding power.