How about baggy shorts, like the ones some mountain bikers wear? Or skorts? I don't have any experience with which hang up and which don't but maybe that's an option? I used to ride around in plain old denim shorts and a tank and that was ok but I've found having a chamois is a good thing. I prefer the lycra shorts now that I'm used to them. I also enjoy the wicking and quick drying action of cycling jerseys and bras though I could easily live without those. Ok, sport bras are ESSENTIAL!!!!

As for answers to your questions:
1) Did any of you have to get over your fear of lycra?
Yes. I confess I'm in a decent BMI range but I have a TERRIBLE body image. I was so busy getting over the diaper feeling of the chamois that I forgot to worry about the rest. I just wear one of my larger, longer jerseys on my "fat" days to cover my hips and belly. The older I get the less I worry about what other people think about my body. I've come to realize most of us have hang-ups and that what we don't like about ourselves is magnified to us but others could care less and would tend to see our good stuff, which we all have lots of if we'd just be brave enough to look for it.
2) Did any of you start out as a Big Girl when riding a bike, and what kind of provisions did you make to make sure that your training went well and that you were comfortable?
See #1.

This is getting kind of long...
Well, have fun and remember that lots of us started as out of shape riders, you aren't alone in that by far and you aren't alone in your worries. Also remember you are out riding for you, let others think what they want (though I think you'd find they aren't thinking about you negatively unless you've been mean to them).