This is what I think happened. Your organic base was an amine, probably an aniline in which someone worked out conditions to form an insoluble bisulfate. As sulfuric acid was being added, the bisulfate began to precipitate. When all of the base was consumed, the pH of the solution began to change withe the addition of a "generous amount" of sulfuric acid. I'm assuming the precipitate formed this gummy material in the bottom of the flask.
It did not form any gummy precipitate.
The solution was yellow before the addition. When adding the Sulfuric acid the solution got less and less yellow as the salt precipitated, after a point of adding the sulfuric acid, salt stops precipitating and the reaction rapidly turns into a dark red color, at which point the salt immediately disappears.
Today, basifying the mixture in an Acid/Base extraction allowed the organic substance to be extracted in the alcohol and then turned back into a salt by and stopping sulfuric addition before the solution turns red, then filtered out.
Me and my partners are unsure whether or not we lost some of the organic substance because even in the acid/base extraction, the aqueous layer appeared to have a salty concentration at the bottom. We are not sure if this is just the NaOH(Lye) we used to to basify it, or something else. As far as I am aware the lye should have been soluble in the water.
So, some of the organic substance was recovered in sulfate form the second time, but is it possible that the sulfate form reacted with the sulfuric acid (in the first over addition) to form some salt molecule that could not be fixed with an acid base extraction? Or was that just lye that I was staring at in the bottom of the Aq Layer after attempting to extract the base? The lye was totally dissolved in water before we added it to the RXN to basify it, so I am not sure what that could be at the bottom.