Is it necessary to clench the fingers when saying "Astaghfirullah" three times after prayer, or is it sufficient to say it verbally?

It is not obligatory to clench the fingertips when seeking forgiveness (istighfar). Seeking forgiveness after prayer is a Sunnah (recommended practice), not an obligation, and it is not obligatory to clench the fingers to count anything. The matter is broad in this regard: if you wish to clench your fingers to count, it is preferable, but if you wish to simply utter it with the tongue, it is valid and does not diminish its reward in any way.