Just to rule out photographic chemicals entirely, try setting up a darkroom session with just water, and go through the entire ritual, for the normal amount of time that you spend in the darkroom. This would include all the gloves and tongs and gear, but just shuffle dummy film or paper between trays or tanks for a while. Then see how you react the next day. If your skin still breaks out, then its not the chemicals directly, but something else - environmental, ventillation, latex, excessive prespiration, etc.
If you do the above control test and don't have any adverse reaction, then try repeating the experiment but with only one of your three chemicals; use water for the other two. In time, you should be able to narrow down the source of irritation.
Good luck and keep us posted.
~Joe