What happen to the EMT lady thats been infected?

The venom went into another host.

Her symbiote (named Riot) jumped into the older Asian lady, then into a little blonde girl and then into Carlton Drake towards the end. Riot was the one Venom and Eddie Brock were fighting with.

The 'parasite' doesnt stay with it's previous host. It moves on from one to the next. The same way that Venom left Eddie, ended up in the dog and then to Eddie's ex before making it back to him.

As for what happened to the EMT lady after Riot left her, I'm not sure that ever see. I assume she died because she was an imperfect host and Riot basically ruined her body from the inside. I could be wrong.

You did miss something The symbiote (Riot) changed hosts to the old woman. The EMT lady collapsed when it happened, presumably dead.