Smile while you sing, it lifts your cheek bones and makes a bigger chamber of your face.
To hum through your nose, hum on the letter 'n'.
The "me-me-me" thing also helps direct the sound through your nose.
Stand (or sit) up straight, your abdomen should be held tight, and when you do staccato you abdomen should move and nothing else. Practice this by standing with one hand on your belly while you sing, you should feel your abdominal muscles working.
Practice inhaling, your belly should distend (most women hold their tummies in when they breath, this doesn't work for singing). Practice exhaling, let out your breath for 10 counts, 12, 15, ... keep lengthening the amount of time you are exhaling.
Move your mouth. Practice singing "oo-eee-ooo-eee-ooo-eee" one one note and exaggerate the movement of your lips with each vowel change.
Open your mouth. Either stand in front of a mirror, or lay one hand gently on the side of your face so you can feel what your mouth is doing.
Also loosen your face before you start. Blow air through your loose lips the make a bpbpbpbpbp sound.
Find a choir/chorale group that has some vocal lessons before practice. You'll get free lessons, make new friends, and learn some new music.
If you do go the private instructor route, find one that will let you share lessons with one or two people at your same level and split the cost.