As someone who is a victim of credit card back in 1990, Yes you should file a police report immediately if not sooner.
Because you are now playing a game under Federal law rules you don't understand. Wait too long, to protests, usually two months, and you lose all protections under Federal law. If you ask me is that logical way for our congress to protect the consumer and I would have to say from bitter experience no. But still its how our congress did it in the 1970's and 1980's in terms of writing the fair credit reporting law and other legislation.
I was larely no where in 1990 in terms of having rights, but our OP is in far better shape now that our congress has amended those laws to give better protections to those that are victims of identity theft. But our OP can't even begin restoring his rights until he files a police report. In 1990 it took me four years to restore my former flaw free credit rating, as my enemy was not really the companies the identity theft defrauded, and instead was the credit reporting agencies, powers onto themselves, that will report any bogus information as gospel truth to any of your future creditors. I had to write enough certified letter chit to play 52 card pick up in 1990, and now one police report, will finally put the fear of god into credit reporting agencies who keep reporting bogus information, as it gives you the right to sue them.
One might think the credit card industry would aggressively fight for the honest consumer and go after the identity thieves, but instead the credit card industry and all their lax rules, are far more interested in making even a fraudulent credit cards are even gooder than gold. As credit card companies they have special funds to simply write off the comparatively small losses from identity thieves rather than require the merchant to be any part of the security process.
Maybe end of rant OP, but if you don't fight for your rights, no one else will.
As maybe, even today, it may to have a horrible credit report, then when a identity thief steals your identity, they will be turned down when they try to defraud you.
As even Identity thieves can't tell the difference between you and Bill Gates. As I can still remember my shock in 1990, when I contacted credit reporting agencies and discovered I had so many credit cards I never applied for. And they were already maxed out.