Contrary to popular belief, all DT and WF lines do not taper all the way to the end. They have a section of level line at the end called the TIP
that allows for fly fishers to repeatedly tie leaders to the end of the fly line without getting back to the taper of the line. The length of this level tip of line varies from manufacturer to manufacturer. Some manufacturers have a very short tip and other manufacturers have a long tip. Usually they are 6-12 inches long.
So cutting back on the tip of a fly line by 6 or so inches will not
necessarily affect the taper of the fly line. What it will do will make the fly line actually better able to turn over the leader because it brings the taper section of the line closer to the beginning of the leader. Cutting most of the tip section off and nail knotting a stout section of mono to a short tip section for a loop to loop connection to the leader optimizes leader turnover.
In fact if you regularly make short casts and have difficulty turning over the leader, cut the tip section off and you will be able to turn those short cast leaders over more easily. The leader will turn over even more easily if you cut further into the front taper. So cutting off some of the taper is not always a bad thing. It all depends on what you want to do.
Note: Some fly lines like Scientific Anglers lines have only a very short tip section which should not be cut off unless you want to cut off a section of the front taper.
Do an internet search for the profile or your fly line to see if you cut off any of the taper.
Regarding the sinking tip. The cause is probably not water seeping into the core. Leon Chandler of Cortland line addressed this issue long ago. Cortland tested how much water entered the braided core of fly lines many years ago and found that very little seeped in and not enough to sink the tip.
I would clean and prep the tip with line treatment.
If the line is clean and treated and still sinks, the fly line you are using is probably not a density compensated fly line. It has been made with older technology.
A floating fly line floats because of the material between the fly line core and coating is made with a material that has micro bubbles. This layer gets thinner as the line tapers. The tip contains the thinnest layer of the material that floats the line.
There are density compensated floating and sinking lines to float the tip evenly with the body or to sink the tip evenly with the body. In density compensated floating fly lines, more bubbles are added to the material as the line tapers, and in sinking lines more tungsten or lead.
With older technology in the cheaper fly lies, there is no density compensation; and the tips sink, because there are not enough bubbles between the core and coating to float the line.