I learned back in my school days that the vowel is the one you hang onto for long notes. How you keep them from sounding screechy, as you put it is by supporting your breath from your diaphragm, and opening your mouth wide, relaxing your jaw and throat muscles. Warm-up exercises will help here. That's part of the reason singers go through all those scales before the show.
If that still doesn't work, try changing the key of the song. It might be too high for your voice to manage effectively. Perhaps as much as say from G to E, or D maybe. Hope this helps.

Jim