On common mistreatment of harmonizers is forgetting that you are INCREASING the signal by as many as 4 times. It's just like adding 3 or 4 people to the "live mix" so naturally, the possibility of feedback is there.
First of all - harmonies should NOT be 4 times louder than lead parts, so ....... SING QUIETER when you kick in the harmony! The total blend of voices should be proportional to the rest of the lead. The harmonies are there to enhance, NOT to overpower.
Another way to deal with it is to back off the mic when singing harmony. Then lean IN when doing lead. This will help the balance issue.
Remember ..... you are now the vocal "ARRANGER" and you have to be aware of the many changes in the orchestration when you add voices. It's not as simple as pushing the button and "VOILA" - Manhatten Transfer appears!
Work on the blend, and the feedback will go away.