Battlefield 1942.
every level allows you to be a sniper. (how you use it is up to you.)
It is my favorite positoin to play in this game for exactly the reason you stated. I can't run and play assualt with these "jumping bunnies" (but i am getting better at following the bouncing ball. lol

) battlefield allows me to stick and move. (rather it encourages you to move by showing the guy you killed where you were.)
you can also call in artillery strikes.
PLUS you could have 14 (or more) CLANIMO buddies backing you (YES YOU) up!
plus there are all kinds of other toys to play with when you get tired of sniping. planes, ships,subs, jeeps,tanks