March is generally the last month of pleasant warmth before the heat begins to crank up
Thing is, no matter how hot it gets in Dubai, you'll be inside with air con (on to the max!) most of the time, so actually, I'd advise layers. Cardigans, light jumpers - and hats scarves and gloves for the cinemas which for some reason are like being in the arctic!
In the evenings when it's not June/July/aug/sep you can generally sit outside in the evenings and it's very very pleasant warmth- dry heat- so no layers required
Bear in mind when in malls, public places etc, although not "insisted" you may get strange looks if you walk around in skimpy shorts and vest tops, especially if you're going to be a resident, so therefore should know better. Keep a pashmina in your bag for quick covering ups if you need to
You can't walk anywhere in dxb, so wearing heels all day isn't a problem 
Have an amazing time. If you need friends out there / advice, there is an expat forum on MN that could help
V jealous. I moved back from Dubai a few years ago and miss it terribly. Send the sandpit my love!