The "doughut" or "coffee ring" is a function of surface tension during drying. The link points to a study of additives during printing that might help -- it was for antibodies. A problem with proteins is that additives suitable for some proteins might not be suitable for others.
Another solution is to dry the printed surfaces slowly, i.e., in a humid chamber, 45% humidity is frequently recommended.
Is it possible for you to alter the nature of the surface you are printing on? This isn't always possible depending on the substrate, and/or what you will be doing with the arrays. There are many treatments to alter the surface hydrophobicity/hydrophilicity.
Regarding humidity, some arrays I printed were done inside a very high humidity chamber, including placing the array printer inside that chamber. This helped even out the drying speed literally from the first moment of printing, giving a chance for the material to adhere to the surface. You must keep the humidity below the point of condensation during the printing, and then carefully and slowly drying from the high humidity condition.