... is it safe to ship it in just the padded case or should I go find some kind of box to put it in?
You mention the case is padded, but not if it's a hard or soft shell. If it's a hard case it's good to go, ship it. If it's a soft case, then you'll need more than PopTart boxes to protect it. Soft cases are good for cosmetic protection only. They offer little if any protection for the structure. One good drop could break a tuning beg, or God forbid, the neck. Double layer corrugated minimum. Also, loosen the strings before storing and/or shipping. Less chance of warping the neck.
Or you could just ship it as you received it and cross your fingers.
GL