Q: How would it be best to try to resolve this as cheaply as possible?
A: DNA tests are cheap these days! For my genealogy clients who are adoptees, I recommend using FamilyTreeDNA. The specific test most taken is an Autosomal test. Very straight forward. The test costs $99 at that site. https://www.familytreedna.com/
There are of course other DNA sites, but this one is my preferred 'go-to' site.
If you want to learn more about DNA testing and how it might help to resolve your case you might read these sites:
FamilyTreeDNA FAQ page: https://www.familytreedna.com/why-ftdna.aspx
Your Genetic Genealogist (and a very good genetic genealogist to boot): http://www.yourgeneticgenealogist.com/
If you are an adoptee I would encourage you to do an autosomal test at the three big DNA sites.
Image courtesy of dream designs at FreeDigitalPhotos.net, Photo by dream designs. Published on 15 September 2014; Stock photo - Image ID: 100288222