Drive if you can (i.e. is your car appropriate to drive big distances in remote areas, and can you afford it). Take your time if you do drive cause there are plenty of things to do and see along the way. You may not get the chance to return to many of the places you drive though so don’t make the mistake of passing them by aiming solely for covering max km’s in a day. You will no doubt have some great stories to tell the folks back home by the end of the journey (take a stack of photos or nobody will believe you!).
Not sure about Broome, but in most other Kimberly towns you are going to pay up big for second hand cars (especially if you are after a 4WD).
If you do chose to leave as late as April, expect there to already be heaps of hopefuls up there putting in the hard yards.