I agree with Nickel. The standard expression one would use for "I'm not religious" is "δεν είμαι θρήσκος / δεν είμαι θρήσκα".
Definitely not "θρησκευτικός", this is never used as a noun and never for people.
As for the word "πνευματικός", when used as a noun it means "confessor" (more specifically, a priest or monk who is one's personal confessor and religious mentor). It is not used much as an adjective for people and rarely in such a context as the one you present here, so it would not be understood. I would omit it entirely.
I agree with Nickel's suggestion and I will also attempt a more loose rendering:
Δεν είμαι θρήσκος, αλλά έχω θρησκευτικές ανησυχίες. Με απασχολούν πολύ τα θεολογικά ζητήματα. Πιστεύω ότι μάλλον υπάρχει Θεός, αλλά δεν ακολουθώ κανένα θρησκευτικό δόγμα.
(Note here that "confession" in this case would be "ομολογία" but it is not frequently used and might also not be immediately understood by all).