what are the specs on this "other" motor... is this some sort of peltier or water cooling setup?
the 37xx series are intended for switch-mode power supplies, that is why their current rating is so high, but you can certainly use them to switch smaller loads!
use whatever nfet you have handy, the 3708 is ok, since it offers a good RdsON at logic-level (+5v) control voltages. The IRFU3711 is physically smaller, and is rated at over 100amps at 20v
just visit the parametric search at International Rectifier and punch in your requirements, it will recommend a fitting switch for you.