I tried adjusting it when I put on a 11-28 - not really to my satisfaction though I would have got away with it, so got a Tiagra GS rear derailleur off the fleabay instead...
AFAIK Shimano road rear mechs of the same model/series have exactly the same large cog capacity (because they have identical parallelograms) be it a SS or GS (which have different cage lengths, which determine chain wrap capacity), so most probably your Tiagra GS won't help you if a Tiagra SS is no good. Mtb rear mechs however generally have longer paralleograms, which will allow for larger largest rear sprocket.
Though I think it is unusual, although not impossible, for a Shimano road mech, which officially has 27 tooth max rear sprocket capacity, to not work ok with a 28T cog with careful B screw adjustment. But of course that is influenced by, amongst other things, rear dropout and gear hanger geometry.
As I kind of indicated in my earlier post above, the best way to spot the existing large rear sprocket capacity is to examine the clearance between the upper jockey and the largest cog, after allowing for a bit of room for the chain, every 2mm will allow an additional tooth.