You should absolutely go for the new job, but ask if you can put in and finish out 2 weeks at your previous job. Not sure why a potential employer would balk at that; if you're putting in 2 weeks to come to them, you'd likely put in your 2 weeks to move on from them, and that's a good thing.

Not at all. There's nothing wrong with improving your own situation and that is what you've done for yourself accepting the new job. The new job is a much better situation for you.

What you should do if you haven't is let the current employer know you're leaving so they can prepare by advertising to hire someone to replace you and either train someone to take over opening or closing or move someone in to your position that already knows how and let them do it.

Give them time to prepare for your departure. It wouldn't be the best thing to do to not say anything to them at all untll your last day and hit them with it as a surprise they're not prepared for.

Take up the new job, it sounds a lot better.

I always secured another job before quitting.

Don't feel bad for leaving for a better position. They will be sorry to see you go, but they will understand that they can't provide what you need. If they were that concerned, they would have found a way to give you full time hours.

You are not a bad person; you've at least considered the boss's dilemma.

But you have to look after number one, yourself, before you can help anyone else.

If I were you, I'd approach the hiring person at your new job/place and ask for a delay in starting there. And during that delay close/open the store and, important, train someone else to do that. And as soon as that's done report into your new place for work without future interruptions.

And if the new place insists on you reporting for work right away...at least you tried. But you really have to look out for yourself before you can look out for anyone else.

It's great that you don't want to put the old job in a bind, that's how it should be. However the old job shouldn't put you in a bind either. Give and get, right? Also you can't control when opportunity will arise. Also there is always another emergency, another issue, another deadline; the perfect time will never really come. 

The old boss will hate you if they are selfish, or will understand and just be frustrated with their current issue. This is what being the boss is, constant issues. It's time for you to move on and the boss will handle it as the boss is supposed to do. 

There are times in life you need to be selfish. This sounds like one of those times. If you asked for more hours, and were turned down - then the situation is "on them.."

You have to think of yourself. However, perhaps you could explain the situation to the new employer and see if they could wait an additional five days before you begin work. I got a job in California and lived in Pennsylvania. I asked for two weeks and they said no. I flew back, bought an old truck, packed and drove back out in seven days.

Assuming you gave your current employer 2 weeks notice and agreed with your new employer a starting date there is no reason to upset the schedule.  I’m sure moving on will cause a burden for your ex employer but working 

15 -22 indicates you will be replaced with the remaining employees.  I would not try and delay your new starting date.  More importantly their schedule should not be disrupted to accommodate the ex employer.