#1
I'm fairly new to guitar, and I'm not learning individual chords.

I'm finding it difficult to keep my fingers in one position, and as I result, it's hard for me to play songs that don't use individual notes (read - most songs).

Has anyone got advice on how to get better at this?
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#2
practice

You can do it, just not very well or very quickly.
The more you practice the better you'll get at it, you just have to be patient and disciplined. Chords are essential, they're a guitarist's bread and butter and an unavoidable part of learning to play the istrument.

On another note, how exactly can it be "difficult" to keep your fingers in one position? Your fingers do what you tell them to do, if you don't want them to move then don't move them. Ultimately that's a lot of what playing guitar boils down to, learning to control your fingers and getting them to do what you want them to do. Keeping them in one place is the easy part, it's getting them to move exactly where you want them to which is hard.
Actually called Mark!

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#3
Try strength excercises. Perhaps just holding a chord for as long as possible.
Stress ball or hand weight excercises to strengthen your fret hand.
chromatic scale runs help as well. ie. from top fret.1234(fingers)down all the strings.

E-1-2-3-4
A-1-2-3-4
D-1-2-3-4
G-1-2-3-4
B-1-2-3-4
e-1-2-3-4

Hope this helps bud.