I seriously wouldn't use peat -- the problem is that once it dries out it is real hard to re-wet. Thus it will blow away and/or create a water-repelling layer atop your lawn soil.
If you want to do something like that, I'd suggest fine-textured compost (probably commercially purchased). That will give you a bit more organic matter for the soil without messing up its moisture-accepting ability.
But honestly if your soil is that sandy, topdressing is not going to do any meaningful good and might even make the problem worse by encouraging shallow roots. Sorry, but, you know.
If you really want a more moisture-retentive drought-hardy lawn, your best bet is to grit your teeth and TILL IT ALL UP, tilling in a substantial amount of compost and also something with more long-lasting organic structure such as wood shavings or fine tree chippings (preferably composted -- if it's not composted you'll have to fertilize heavily with an N-heavy fertilizer for the first year, multiple times, using your eyeballs to determine when more is needed). Then level the seedbed and reseed with a new lawn.
But if you are in a hot climate, this will not work forever, eventually (possibly in just a few yrs, depending where you are) the organic matter will have disappeared and you'll be back to nearly where you started.
Sometimes you just have to kind of accept that the prototypical suburban/English style carpet of green lawn is not going to happen in all locations.
Good luck,
Pat