Yes, but it will be complex. There are videos online showing how experts can physically remove the motherboard, clean it, and insert it into a special device. All of your information will be there. However it isn’t free, so decide if you really need it (its not too expensive though).
It is very possible you purchased an after market LCD. I have done this several times, and they were hit or miss. I suggest looking at the reviews of those retailers you are buying your item from. They are so cheap, I recommend buying two LCDs- with one as a backup. Finally, you may want to use compressed air to clean the “lego” like connectors when you plug them back in. They may be dirty, or not properly connected. It should be a satisfying “snap” when they go together properly. Finally, make sure you aren’t pinching the ribbon as you are putting the item back together. I hope these trouble shooting tips help!