It is possible, but not guaranteed, that a Shimano 105 medium cage rear mech (presumably what you have) will take a 34T sprocket - as you say there are reports of people trying this successfully.
Another option however is to change your chainset, to either a MTB double (say 40/28 - which will obviously be a big difference) or some of the super-compact road cranksets that are now available (e.g 48/32)
Replacing an individual sprocket may not be possible depending on your cassette, the larger sprockets are often joined together. I think the lock nut also screws into the smallest sprocket so you may not be able to replace that either. In any case Shimano absolutely do make 11-34T cassettes - for their mountain bike groupsets.