You will most decidedly need a heatsink for any linear regulator solution here. You're talking about dropping (12-5)*2=14W on the regulators (to only deliver 10W to the load). Even with a reasonably good Theta_JA of 3 degC/W, you'll see a temp rise of 42C (108F) above ambient!
If you do want to use a linear solution, the venerable LM323 is probably one of the few devices you'll be able to suitable heatsink. Note that on this device, Theta_JC is 2 degC/W, which means that even with a "perfect" heatsink, you'd still get a 28C (82.4) temp rise.
This is enough power that you might want to find yourself a switch mode power supply. There are some solutions out there for 12V to 5V capable of serveral amps. You may have luck using devices designed for automotive use.
You can try building one yourself (the topology you probably want is the "buck" topology), but without some serious electronics experience, this is likely to be more frustrating than successful.