Kitchen sink not set into worktop

My current stainless steel sink with drainer is the standard type set into a worktop and I have endless problems with water getting into the worktop and down the wall. The worktop has been replaced once and has again rotted. Sealant replaced, but it keeps failing and so it goes on. The sink is on top of a 600mm unit and the draining board is set into the worktop over a washing machine. (My kitchen is miniscule.)
It also looks ridiculous having a tiny edge of worktop to the back and on one side of the sink. I should like to replace it with a sink with draining board that sits on top of the unit and which fits flush to the back wall and the the right hand side too. so the tiling will sit over the sink rather than over the worktop. Even better would be to find a sink which rise up at the back and on the right sides with the drainer on the left. The drainer will still need to have the washing machine under it.
is there any way this can be done, please? And does anyone know of a brand/model of sink that would do the job (I have looked extensively)?
Thank you for any help.

Firstly, all cuts should have been sealed. This reduces the risk of waterlogging. Secondly, the sink, if fitted properly should not leak into the worktop. This said, if you want a sit on then I like the ceramic ones. Focus have one for about £120 at the moment. It will still have to be sealed on fitting as will the worktop.

Yes, as fp said, you are supposed to seal the cut edges of worktops in order to prevent water absorbtion, beause if you don't the wood particals of which the board is made will eventually swell up and rot.

What you usually seal them with is PVA woodworking adhesive, it's that white, water soluble glue that you use for sticking pieces of wood together.

Paint a liberal, undiluted coat of PVA adhesive onto the cut surface, it soaks in and becomes transparant when it drys.