I respectfully disagree with bpoletti when he says that the "absolute BEST a LINN can sound is nowhere close to modern tables in that price range." I suggest that you find a competent Linn dealer to help replace the springs and grommets, tune the suspension, replace the drive belt, assess the cartridge and replace the bearing oil. This should cost only a few hundred dollars if the cartridge is OK. A Denon DL103R is a great cartridge for a reasonable price - around $300. You would end up well under $1,000 and almost certainly have a turntable that competes very nicely with modern turntables in the $3-4K price range.
Should you choose to upgrade, you could get a Cirkus bearing kit ($750) and Kore subchassis ($1,200). These are worthwhile upgrades as is the Lingo 3 power supply, though at $1,800, that would push to the upper limit of your budget.
If you can find a dealer to demonstrate these upgrades, I suggest that you hear and assess these for yourself. Rely on your own experience but don't automatically give up on the Linn because of its age. The LP12 can be maintained for a very modest cost and upgraded at varying price points. Based on my experience, the Cirkus, Kore and Lingo 3 are well worthwhile. At the highest upgrade level, an LP12 competes with turntables at very high price points.
Thomas O'Keefe at Overture Audio in Ann Arbor, Michigan, and Rick Duplisea of The Audio Alternative in Fort Collins, Colorado, are both excellent dealers and Linn authorities whom you could contact for expert help with your LP12.