I don't think you need to move your weight forward much when standing to climb - i know I keep my weight back, with the nose of the saddle hitting my butt on each pedal stroke. This brings the muscles on the back of your legs into it. As long as I don't get crazy (the temptation when standing is to accelerate and go fast) I can climb like that 5 - 10 minutes and use that to let my normal cycling muscles recover on a long climb. Although you won't get as much of a benefit if your feet aren't clipped in.
How much higher are the handle bars on this bike than on your old bike? Are they higher than the saddle? That might be part of the problem.
I still think the shoes might be part of the problem. I can't imagine trying to stand on a climb wearing crocs! All you could do is push down, for everything else I'd be expecting my feet to slip. At the very least, try toe clips.
Finally, please don't change a bike fit that works otherwise because of how it feels when standing! You will be seated far more than standing, so you really want to optimize for that (particularly with a bad back). Did you have a bike fit done? I've seen REI do a good job with this, but they completely screwed up the fit on my ex-wife's bike so I've seen them get this pretty wrong, too.