Citations to make settlements and other citations or notices to such nonresident executors and administrators, and all writs or legal process, including executions on decrees and judgments, may be served by depositing a copy in a sealed envelope, postage prepaid, in a post office at or near the court or office issuing such process, such envelope to be plainly addressed and directed to such nonresident executor or administrator at the place and by the name furnished by him upon the filing of his application for letters, or thereafter as provided in section 43-2-190 and, when the sureties, or either of them, reside in the state, and their place of residence is known, by the service of a copy of such writ or process upon one of such sureties. When service is had by depositing a copy of the writ or process in the post office only and not accompanied by service upon a surety, such service shall not take effect until the expiration of 10 days from the date of mailing such copy. If the post-office address of such nonresident executor or administrator is not furnished as required by section 43-2-190, service on one of his sureties is sufficient. If such post-office address is not so furnished and if there be no resident surety, notice of such writ or process may be given by publication in a newspaper published in the county wherein the proceedings are had for two consecutive weeks, and a copy thereof must be sent by mail to such nonresident executor or administrator, if his post-office address can be ascertained; and, if in such case there be no newspaper published in the county, the notice may be given by posting a copy thereof at the door of the courthouse for 15 days. The sheriff or other officer executing the writ or process must make return thereof according to the facts.