Solfege and sight singing are two different, but related activities. Solfege (either fixed or movable) is the practice of assigning specific syllables to either pitches (fixed) or scale degrees (movable). By way of example in fixed solfege - do re mi fa so la ti do is the C major scale, in movable it's ANY major scale. European schools seem to favor fixed, American movable.
Sight singing is reading a piece of music and singing what you are reading instead of playing it on your instrument. You can do that with or without using solfege syllables.
"Improve your ear" - well, ANY work you do to connect what you are seeing on the printed page to an aural experience is good for your ear. ANY work you can do to connect a sound in your head to a readily identifiable pitch is good for your ear.
And I do agree whole heartedly with BRAINRUST's suggestions that time away from the instrument is best spent working on your ears, I would like to add that a haphazard approach to ear training may not get the results as quickly or build as strong a foundation as a more studied approach. Teachers are a good thing. They're not going to let you blow through something that you almost have down, they're going to keep the exercise moving you forward on the good solid foundation you've built before.
If you just absolutley can't find a teacher, there are any number of courses out there (Gary Willis, Berklee, Dick Grove etc etc) and just about every community college in the US with some music program will have courses in sight singing and ear training. My personal experience has been that it is easier to make progress one on one with a teacher than in a class situation. But it's like running a marathon - you don't just buy a pair of shoes and sign up. You work slowly and steadily to build up your endurance and stamina and mental state, you train for the next level, not the end of the race.
