I've been doing strength training on a fairly regular basis, along with cardio when I can (darn machine broke), and I can tell from how I feel that my muscles are getting stronger. The thing is, the flabbiest parts look flabbier than ever now - especially arms and tummy. I mean, I can see muscles developing, but the biggest fat deposits really hang there now! Is this normal? Is this a stretched skin issue, or will these areas shrink up as I lose more weight? What should I do to take care of this?

It's probably a combination of skin and fat. It will likely get better to some degree, but there may always be some lose skin to deal with as well. This is probably something that is best evaluated by an appropriate medical professional.

I have heard that when your fat cells shrink, it is not evenly and so it take time for your body to adjust. This is why people will still lose inches even when their weight loss is stalled. I have noticed on myself that the flabby areas are loose and squishier than before. I imagine that the body just need time to rearrange things on a cellular level.

i've found a pattern of tight fat, loose fat, fat leaves (inches lost), things tighten up again - and then repeat.