No projector purchase should be considered an 'urgent' matter. You should always slow down and talk about what you are trying to achieve and what your situation is.
If you have the budget for a better projector, like you do, then I can only assume you also have a room which is appropriate for a better projector. This means that you have a room with dark paint, dark carpet, dark ceiling, and proper light control in general. If not, then I would encourage you to spend some of your budget making the room a more hospitable environment for your projector.
Do you have a desire to purchase a anamorphic lens? If not, then the 6020 doesn't make as much sense as the 5020 which is basically identical, but costs less overall.
Finally, JVC will deliver a better image - period. But, only for 2D. The JVC projectors apparently are still struggling with their 3D performance issues.
There is also the Sony VPL-HW50ES which is around $3,500 or so and fits your budget nicely and is a VERY strong projector for the money. Very well rounded.
Sony VPL-HW50ES Projector Home Projector Review - Overview
Beyond this site (which should have all reviews checked out) you should look at ProjectorReviews.com and ProjectorCentral.com for up-to-date reviews of models.
I always find the JVC models to provide some of the best black levels with great reliability, but the Sony models have been stepping up recently with overall quality and balanced quality for the money.