Look at the point & shoot cameras that are rated for water use.
Up to ten feet under is plenty.
Means they have seals that will keep dust out too.
The models out in the last year have some pretty impressive photo results, and in quite varying conditions.
Alternately, if a more appropriate camera isn't in the budget (sigh), get the camera shop to show you how to safely blow it off with air. Don't know if canned air (clean air) or a rubber squeeze bulb (which may blast air/dust combo into mechanisms) is best for P&S. I did Africa, including the Namib, Etosha and the Kalahari using a rubber air bulb frequently, but that was with a Nikon FM2 (mechanical) and a F4 (which is pretty well sealed), so that success may not apply to current mechanisms.
Also, frequently wipe down the outside with a micro-fibre cloth, and then place the camera in zip-lock style plastic bag between photos. Edit:
I'd started out using the plastic bag last year, but reverted to just wiping and wrapping the camera in the micro-fibre cloth and sticking it in my pocket.
I had a Sony P&S last year, and it got dust, but apart from the auto-lens cover need the occasional coaxing to open, it worked fine.
Until last week when there's suddenly a huge piece of dust inside on the sensor where I can't get at it.