This may be strange advice but I’ve found that if I can get my name and the reason I’m calling out of the way, then the rest is easy. Something I’ve tried is recording on an iPad (voice memo) and then playing the recording when someone picks up. (Most times I’m completely relaxed and fluent while recording, or I can keep recording until I get one that I like.) Then I continue the call like normal. If the sound is different people just chalk it up to speakerphone or Bluetooth.