Proper Technique for a Pushup

Hello everyone!

I just recently joined FB and I love it. I, unfortunately, had to switch from doing the FB Bodyweight to FB Low Impact cause I couldn't handle working out that intensely and waking up at 6 AM every morning for school (lol). However, I'm still running into a snag---the pushup!

A bit about me: 25 (26 in June), female, mostly sedentary lifestyle (medical school). I've never had issues with lower body workouts, but I've come to realize I can't do ONE normal pushup--knees or on feet. My question is this: do I have to go all the way down (i.e. chest touching the floor)?

When I started FB Bodyweight (got about 2 weeks in before having to stop) I could only go about 1/4 of the way down. Now that I'm starting FB Low Impact I can go about 1/2 of the way down and do about 20+ of these. This goes for either on my knees or feet. I've read I could start doing wall push ups, but they are much to easy for me. I can't get a workout bench due to my location.

I was wondering if doing what I'm doing (slowly going lower and lower) is good enough or if someone has a better alternative for me to do?