A friend of mine is a master tech for Honda/Acura. I asked him what to do when I was experiencing the water in the trunk issue. He told me (I **** you not) that the actual dealer fix, is simply replacing the taillight gaskets with oem gaskets and drilling a small hole in the bottom of each light section. So I did this, plus cleaned the tail lights and mating surfaces as best as I could and sealed the top/sides of my taillights with black RTV (I left the bottom unsealed plus drilled, so any water that possibly makes it in, can easily drain out the bottom of the lense and not into the trunk).

I did this about a year ago and haven't had an issue since.
I also replace the front windshield plastic cover cowl/gasket and bent the A/C air vent intake tab back at the same time, because that is another common source of interior water leaks.

 

your tallights are probably fine, an old worn gasket is usually the source of the leak--which makes sense why you think both taillights are leaking if you re-used your gasket on the new ones. look up the gaskets on ebay. there is a shop that sells thicker than OEM gaskets and they work great. something like $20 bucks or so but well worth it.

https://www.ebay.com/itm/DMT-HONDA-94-1994-95-1995-96-1996-Acura-Integra-2dr-Taillight-Gasket-Gaskets/291130346838?fits=Model%3AIntegra&epid=26010721978 &hash=item43c8b88d56:g:EpUAAOSwWTRWvTUC&vxp=mtr

i also had water in my taillights and could not get it out so had to drill a tiny hole as well. works great for keeping taillights condensation free.