I agree with Hank on this one. The loan officer should have called you and told you of this increase. Unfortunately, sometimes the loan officer doesn't even know what's going on with the file after it goes to processing. Depending on who your lender was, you might want to talk to a manager and find out why the increase.
Points can only be charged if the rate went up. You may have requested the rate lock, but sometimes, loan officers will float the rate in hopes that the rate will drop and they can get a little something on the back end called yield spread. Back in the day, the loan officer was allowed to put this in their pocket. Since 2010, that stopped. So, the loan officer can keep the fee in their bank to use on the next person who wants a competitive rate.
In order for $6000 to be added in points, you had to have been offered a rate of 2.5% at one time and it jumped to 3.5%. Also, the percentage depends on the loan amount and we don't know what that is.
Another thing that causes "points" to increase could be a miscalculation of mortgage insurance if going FHA or taxes and insurance due extra at closing among other fees (Attorney, appraisal inspection, flood insurance, days added to closing. If your debt to income ratio was tight, then the lender cannot increase your rate to get that yield spread to cover your increase in fees Because you won't qualify with a higher interest rate, you end up paying points instead.
I would find out why first before killing the deal. The seller may not allow you to go out further. Whenever anything is changed on the GFE that affects the APR, the lender is required to give you 3 business days to think about it before moving forward (Rodney, correct me if I'm wrong). Use that time to find out what happened and get your Realtor involved. Lenders rely on business from them and they have some leverage if needed.