Your hair sounds similar to mine, fine and thin with a tendency to wave. A razor cut will certainly emphasize the wave. I have had razor cuts with very poor results, they make my fine hair look whispy and fly-away. Hair like yours is probably best with a chin length blunt cut, either in an A-line bob or all one length. Shoulder length if you want to make a ponytail for exercising.

Without being able to blow dry, styling can be limiting with fine hair. It's hard to get any volume though you could let it air dry completely and then use a bit of hairspray and velcro rollers. You could leave it one length and emphasize the wave with a curl enhancing product. My daughter has curly fine hair and likes Redken Fresh Curls Spin Control, a leave in product that makes her curls defined but still soft, no frizz.

Here is a searchable hairdo database. As for what to do now, let it grow and have the whispy ends trimmed up as you go. been there, done that. I won't let my new hairdresser go near my hair with a razor!

Here is a link that might be useful: Hairdos

For truly minimal care and no blow-dry, either go short enough to put some mousse in and air-dry, or go long enough to wear in a pony tail all the time. Anything in-between needs more grooming for hair that's got any frizziness.

My hair sounds a lot like yours; fine, natural wave, straight sides, etc. Razor or Texturizing this type of hair sometimes works well, but most times will make the hair lose volume, curl and make it look sloppy "choppy" especially if you're not going to style it with heat. I can't remember one good razor cut I've ever received. My hair has always looked best and neatest with the style lasting the longest, when I'm given a precision cut of soft long layers, especially around the face (cheek bone, chin) so that it can be worn wavy or curly with minimal texturizing. Hairdressers just love to pull that razor out though!