Big tires are rubbing in certain conditions so how can I fix my rear wheel play issues?
Unfortunately 28mm nominal tire width can be too larger for many modern road bikes. Note that actual measured with depends on many factors including: internal rim width, tire construction, tire profile, tire age and wear, inflation pressure. As such, one combination of tires and wheels can work while another may fail if you have tight clearances.

It is also impossible to build a bicycle and wheel set without flex.  As such, there needs to be a minimum safe tire clearance. Just because the wheel spins freely in the stand does not mean there will be appropriate clearance for actually riding the bike, especially for heavier/stronger riders.  I would suggest assessing how much clearance is available with 25 mm tires (perhaps(typical max clearance on many road bikes).  If the clearance is already tight (e.g., less than 1 cm) then going up to 28 could be a stretch. 

The more recent version of the tarmac (e.g., disc version) can clear 28 mm, but I doubt older rim brake versions can.  It is best to ask the Specialized Help Desk to get an official answer. 

Question of Hub Play

All wheels, will have some side-to-side flex so it is unclear whether or not there is a problem with the hub. Typically, if the bearings have developed play the wheel moves side-to-side with a very light touch, if more force is required it is likely wheel flex.  5-7000 miles is not a lot of mileage, but this also depends on the conditions they were ridden under.