Go ahead and apply....the worst that can happen is you don't like it and quit.
Seriously though, you say early morning loader, that would suggest either preload (the loading of 3-5 package delivery trucks) or outbound trailer loading.
Either one involves the occasional lifting of over 70 pound boxes which you are entitled to and encouraged to ask for help. Most of the packages are well within the range of easy handling.
Regarding insurance, it's been 25 years since I was a part-timer so things have changed. I BELIEVE (that means I'm not sure) that the company carrys your insurance after a certain number of days. There is also, once again I BELIEVE, a 2 year progression to make top salary. Be patient....there will be numerous persons more knowledgeable than myself that will give accurate info on this message board.
In your situation I would think an early morning part-time job would be ideal. Sure, you may be sore until you become used to it but think of it as an early morning workout that you get paid for. It's also only for 3-5 hours each day, so you can probably handle it. Good Luck.