For me, three things worked to stay motivated and study well while beginning my studying during a busy academic time (and then studying for close to a year): tracking success by hitting the goal time for that day/week (not by question success/failure), making studying LSAT something cozy (I ate strawberries and drank hot cocoa for the first 3 months at least), and deciding that LSAT questions are fun.

Basing it on time means you won't get discouraged by any initial failures/perceived lack of progress, making it cozy helps to prevent you from associating the LSAT with inhospitable/anxiety inducing conditions that could cause you to avoid studying or performing poorly on test day, and deciding it's fun means you'll look insane to your friends because you reach for your study materials whenever you can! :)

If you feel burnt out, take a week or two off and try to remember what initially motivated you to want to take the LSAT and reach your goals—once you feel that you're itching to study again, get back into it!