Is it permissible to pronounce the letter ẓāʾ (ظ) close to zāy (ز) in “Subḥāna Rabbī al-ʿAẓīm,” and the letter dhāl (ذ) close to zāy (ز) in “al-ladhīna” in Al-Fatiha? And does that change the meaning, especially given the suffering from obsessions related to recitation and repeating verses due to difficulty in pronouncing letters?

Waswasah (obsessive whispering/doubts) is among the worst afflictions, so you must avoid it. Do not re-read anything merely because of waswasah. If you doubt whether your recitation is correct or incorrect, do not pay attention to the doubt and continue with your prayer. However, if you are absolutely certain that you made a mistake in a letter, replacing it with another, then you must repeat the word in which you erred. Practice correct pronunciation of letters without affectation. Glorification (Tasbih) is Sunnah according to the majority of scholars.