OK - first up if you searched for "4AGE conversion" on this forum and found nothing something is seriously wrong. Search and you will find.
Second, I think your budget is far too tight and will never be sufficient. But, things you need to consider are EFI fuel lines and tank, electrics/electronics inc loom and ECU, brake upgrade, engine mounts, diff upgrade if RWD and the list goes on.
You could consider a bolt in job such as a 4AC AE82 Corolla. 4AGE physically bolts straight in but you still need to consider the rest.
Going for an older RWD Corolla adds the complication of converting the 4AGE to RWD.
I think you should consider an older Corolla and working with what you've got ie modifying the existing engine and drivetrain. $2000 could potentially see the purchase of the car, engine, suspension and brake modification and result in a reasonably neat package.