say does it cost anything to check the LDS database
No, not a dime, but you need to verify your findings because there are lots of errors in the database due to the fact that the transcribers can't read the old German script or Latin script in catholic churchbooks. To verify, you will have to order the relevant microfilm (the number comes with the dataset) at one of your local LDS Family History Centers. They will contact Utah and ship the film to your location so that you can read and print out the churchbook entries/civil registration records and others yourself. Costs a few bucks, but nothing major.
If you want to trace back your ancestors, I would start at the Social Security Death Index which is also available at the LDS web site I linked. Once you know the SSN and the place it was issued, you can follow that lead to obtain the Naturalization Record of your ancestor. Plenty of information on there. You can also check the web site of Ellis Island and see if you can find your ancestors in one of the ship manifests. Follow the trail to their place of origin, check online databases of the Polish and German state archives, and so on. The Polish ones have already released tons of churchbook scans and other material.
ellisisland.org (sign-up is free)
archiwa.gov.pl
I'm thinking my original ancestors were either the Dabrowskis or Dobrowskis
Or Dombrowski, which might also be a variation of your family name.
I wouldn't understand how that name would change to 'Doberski'
Up until 1878 there were no offical rules how to write your name; the priests for example usually wrote the family names the way how they were pronounced. In genealogy you will very often be forced to trace various variations of your surname. That's normal.