So, like the previous editions of D&D, I didn't care for 5e's poisons. The idea of being poisoned in 5e is little more than bothersome. Poisons should be far more terrifying than they are in D&D's current iteration.
I went back and dug up an old poison system I had written for 2nd ed. and updated it to 5e. I feel like it does a better job of reflecting the wide variety of poisons that exist in reality, and - more importantly - the system adds a much broader array of possible outcomes from being poisoned.
Below is the PDF version. A good DM will work out the details of any poisons characters might encounter before the session begins.