I Have Been In A Relationship And Have Recently Discovered

I have been in a relationship and have recently discovered that my boyfriend and I MIGHT share the same grandfather; it would be my boyfriends great grandfather and is my great great grandfather. There is a family secret that my great great grandfather might have fathered another child who happens to be my boyfriends maternal grandmother. This would basically make my maternal great grandmother and his maternal grandmother half sisters, where they would share the same father, but different mothers.

What kind of cousins would that make us...2nd, 3rd cousin, 4th cousin etc?? and if we had children, would they run the risk of birth defects?

Also, what affordable testing could my boyfriend and i have done to determine if we are related since this is all speculation and all great grandparents in question are deceased and of course DNA testing was not done back then to confirm?