(a) Should any person entitled to the administration of an estate, or to the guardianship of any minor, or person found to be incompetent, fail or neglect to apply to the probate court having jurisdiction, and take out letters of administration and guardianship, within six (6) months after the death of the intestate, or within three (3) months after the settlement of the estate by the administrator, it shall be the duty of the administrator or guardian, provided for in this part immediately to enter upon the administration or guardianship of the estates, as the case may be, first applying to the probate court, for the necessary letters of administration or guardianship.
(b) Should any person entitled to serve as trustee under an inter vivos or testamentary instrument die, decline to serve or resign and if:
(1) There is no individual or corporate successor or substitute trustee designated by the instrument who is willing and qualified to serve;
(2) The instrument does not contain provisions for the appointment of a successor or substitute and no adult income beneficiary or vested remainderman petitions for the appointment of a successor or substitute trustee within three (3) months after there is a vacancy in office or all sooner waive this right; or
(3) The court in its discretion in a proceeding pursuant to § 35-15-414 determines that a small trust should not be terminated but a successor trustee should be appointed;