One way to tell if a lamp is old and needs replaced is to look at the glass near the end of the tube.
If the glass is darkening at either, or both ends, it's definitely past its prime and should be replaced.
Perhaps the smell could also be from a cheap/failing starter? Does the lamp flicker when it turns on?
Many (most?) inexpensive light fixtures have a wall-wart ballast and a starter tucked behind the 'reflector' behind the lamp.
See
'How Fluorescent Lamps Work (Pg. 4)' for how this works.
Don't really know why this would cause that, but it's a thought.
If you want to try replacing the starter, you can find them at your
LFS for $2-$4 for a two pack, or at a hardware store for slightly cheaper. You need the same rating as what you have, so hopefully you've replaced the lamp with the same size/watt rating as the original; the new starters have numbers that tell their type, FS2 (14/15/20W) or FS4 (13/30/40W) are most common for our fixtures.
You can see one of these on that site link, but the ones in our fixtures (usually) don't have the 'can' part, so you have to take it off (easy) and connect it the same way as the old one.
Be sure to tape all the leads so that there can be no shorts. Oh, and I certainly hope you know that you should have the lamp unplugged before taking it apart; I recommend unplugging when the lamp is on to aid in discharging any stored current in the ballast.