Yes, there is increase in sodium absorption levels after rinsing mouth with salt water. This absorption is through oral mucosal lining. Increase in sodium is very little as compared to the daily requirement of sodium.
In the clinical study (link mentioned in question), 5% saline solution (which contains 855 mEq of sodium) was used. Sodium absorbed by various participants was between 2 to 11 mEq, and the minimum requirement of daily sodium dosage is 44 mEq. So, this will pose little risk on metabolic process. And it is possible that on increasing saline concentration of mouth rinse, sodium absorption may also increase. So, 5% Saline solution is recommended for mouth rinsing.
Via: Image of the research article mentioned in question (unable to upload PDF)
Earlier in 1950's various researchers have reported the use of sodium salts of various drugs for sublingual absorption.