Stagger – You're talking about two different problems that can be related, but not always.
Some fly lines float better than others.
If your leader is sunken, it's common for it to pull the tip of your line under as well. This happens to even the best of lines. To help keep your leader floating, put some floatant on your fingertips, and run them down the leader and up the fly line a bit. I prefer not to put floatant on the tippet within a foot or so of the fly, but that's personal preference.
If your line is dirty, the tip often sinks. Keep it clean. I doubt your line has gotten dirty already – you haven't fished it much since I cleaned it. The stuff I cleaned it with contains teflon, and it helps your line float. It wears off after about a day of fishing, but the line isn't necessarily dirty at that point.
The core of most freshwater floating lines is made of a braided material. This allows it to be supple, but still strong. If there is an exposed section of this core, it can wick up water, and sink the line. An unsealed tip of the fly line can do this, as well as breaks, or cracks in the fly line's outer coating. You can easily seal the tip of the fly line by applying a glue to seal it (lots of choices). A break in the outer coating, or cracks due to age (etc) likely means it's time for a new fly line. Your line looked to be in pretty good condition – did it have a welded loop? If not, just put a tiny drop of super glue on the exposed end to seal it, after it dries out for a few days.
Fly lines can, and often do last a long time – especially if you take care of them. Unless there's visible sign of damage, yours is probably fine. You haven't used it enough to wear it out.
All that said, it wouldn't hurt to replace the line. There's some pretty good lines out there that are relatively inexpensive. One of my favorite lines is Hook & Hackle's Hi-Floater line. It's $34, but you can often find it on sale for around $25. There's lots of other reasonably priced lines out there as well.
If you replace your fly line, you could always keep your old line as a spare. Stuff happens, and having a spare handy could save the day. If you have a spare spool for your reel, you could keep it lined up, and ready to go. You could also use your old line for practice casting on the lawn.
