So I used the blue syringe on the tentacle monster, and now it's just some goo, so what then?

It says I can't kill it without a weapon (says you can't wrestle it when you use one of the non weapons), but the gun does nothing to it 

Also, the hospital expedition text is now green as opposed to black, so does that mean I can't go any further?

You need to find and use the sword (yes you need to explore to find it) and kill it that way. After that any time you visit the Hospital, you get a guarenteed Medkit, though the pages will be blank for some glitchy reason. I still have zero idea where you're supposed to find Amber though. Unless I missed some area that I can explore.