When you moved him into the new tank, did you transfer the existing filter (and, more importantly, its filter media) or did you set up a new filter with new filter media? If it's a new filter with new filter media, or if you changed the filter media or ran the existing filter media under tapwater rather than squeezing it in being-discarded aquarium water, then the problem almost certainly lies with the filter going through a fish-in cycle. Following tips from this article
http://www.fishkeeping.co.uk/articles ... ammonia-nitrite-spike.htm should help you address this but large daily water changes will be key - these will reduce the presence of ammonia or nitrite which are toxic in any quantities (and thereby ought to reduce the swimbladder type problems as well as other potential illnesses).
If you transferred the existing filter (or at least its filter media) into the new tank and didn't change the filter media or run it under tapwater, then there may be another reason accounting for the problems. However, without your own test kit to monitor water quality as previously advised, this is impossible to tell - the reason for having your own test kit is to be able to monitor ammonia, nitrite and nitrate regularly as this will fluctuate over the course of a week between weekly water changes and a test is only valid at the date/time it's done.
If you can be more specific about the tank size, water change regime, and also let us know feeding regime, what type of goldfish he is (is he the blackmoor that you mention having previously?) etc, we might be able to get closer to identifying the source of the problem, if it's not the filter cycling; thanks.