Refillable. yes you can add liquid to it until such a time as it is no longer 'fresh, is very different from a rebuildable setup where you can clean it and replace the dirty coils etc.
Cartomizers can be cleaned a bit to extend how long you cn use it for - at least some of them are but otherwise once you have used it enough that gunk from the heating
juice fouls the coil and atomizer its a toss and replace. There are many good video's on You Tube about caring, cleaning and - if you have a rebuildable one - how to rebuild cartomizers.
A burnt taste usually means one of 2 things. Your dry and need more fluid or the fluid has not yet been soaked in enough to reach the coil part of the atomizer (such as a very new one or your cartomizer has reached the point where it needs cleaning/rebuilding or tossing.
There are many different cartomizers and clearomizers,
tanks etc on the market, so you need to know what your getting. For the most refillable does not mean it will last for ever or even more then a few days to a week or so depending on how often it is used.
It simply sounds like your cartomizer, namely the atomizer part of it has gotten dirty and reach end of life - its takes burnt and/or the taste is very off. I would suggest getting a new one and trying that and it should solve your problem. If you find out you have one that can be rebuilt, then you can follow the steps to do that if you are able to and have the right tools, and use the new one while your older one is broken down for rebuilding. If you don't have all the tools, get the parts or just are not that hardware handy and savvy you might find that its realty cheaper to not buy the rebuildable ones and just toss your cartomizer when it get gunked up.