The current one only has a little bit of play in it
Sounds like the cones have slackened off so less tension is on the bearings. Fairly easy to tighten with a pair of cone spanners (or just cheap thin spanners often in flatpack furniture). If you haven't had the hubs serviced periodically throughout their life then you may be in need of new bearings, or worse, new hubs.
but it is rubbing on the brake pads
The wheel is no longer true, or the brakes are badly adjusted.
and is quite unsecure when I go around corners.
Could be caused by the play in the bearings or you could be in need of new (high quality) tires.
It also keep locking up under heavy braking ... It is not under normal braking but when going down hills and I have to stop for something (junction/car)
Are you sure that you are not relying too heavily on the rear brake and not using the front brake enough? It's normally pretty easy to lock up a rear wheel on a road bike.
You may not need a new wheel. I'd advise either doing the work yourself, or give the work to an LBS who won't try and sell you a rear wheel for the extra profit.