I'm having a lot of difficulty keeping my fingers nice and close to the fretboard like they should be. Especially troubling is every time I press down with my ring finger, my pinky comes way high up in the air, and when I'm doing hammer ons and pull offs, my pinky goes underneath the bass. Also slightly happens with my index finger but to a much lesser extent. Any advice or exercises I can do that will fix this? Thanks.

One-finger-per-fret...maybe on the A-string...

Index = 9th fret
Middle = 10th fret
Ring = 11th fret
Pinky = 12th fret

Since you mentioned the Ring/Pinky problem-
Move ONLY the Ring finger up...then down...up,,,then down.
USE YOUR OTHER HAND TO HOLD THE OTHER FINGERS IN-PLACE.
Do this with all the other fingers, too.
Remember, only one finger moves up/down...the rest need to stay stationary!

This is simple but frustrating...so stick with it. Give it a week or so...hopefully, you should see some finger independence
taking hold.

and when I'm doing hammer ons and pull offs, my pinky goes underneath the bass.

...also, be aware of the angle of your fretting hand's wrist. It shouldn't be angled like a guitarist's (which is good for chording).
Sounds like you need to pivot your wrist more towards your body/to the right?

there's a great exercise I do when my pinky starts to flare up...You have to do it at very very slow speeds and slowly work up. The results are immediate if you ask me.

Do a chromatic scale moving up a string each time...So, 1234 on E string, then 1234 on A string, 1234 on D, etc. The key is that you must leave every finger pressed on the string as you move the other one.

So, hit 1234, and when you go to the next 1, the middle, ring, and pinky stay glued to their frets. Then you keep the index on the A string, and move the middle finger, keeping the ring and pinky still glued to their frets. Then lift the ring finger, keeping the index, middle, and pinky stuck on their frets. Then finally move the pinky. Repeat all the way up and down.

Whether you practice this or not, I think part of the fundamental problem is you have too much muscle tension in your arm and hand. The arm needs to be extremely relaxed, almost limp.

Well there's a great excercise on guitarprincipals.com that addresses this. I know it's for guitar, but I think it's applicaple to bass.
It's called the light finger.
Basically, set all your fingers in a position (somewhere towards the middle of the neck), but don't press the frets down.
Just leave them all lightly touching the strings, with no pressure.
Then practice lifting each finger and bringing it back down, without moving the other fingers.
Other than that, try to find out what makes your fingers go nuts and slow it down while trying to keep them relaxed.
Good luck.

I'm having a lot of difficulty keeping my fingers nice and close to the fretboard like they should be. Especially troubling is every time I press down with my ring finger, my pinky comes way high up in the air, and when I'm doing hammer ons and pull offs, my pinky goes underneath the bass. Also slightly happens with my index finger but to a much lesser extent. Any advice or exercises I can do that will fix this? Thanks.

Playing scales and finger independence exercises will eliminate this. When I was beginning back in the early 70's I had the same problem. One thing that helped was playing with only my ring and little finger and only with my index, ring and little finger. Doing the "spider" exercises should also help.

Well there's a great excercise on guitarprincipals.com that addresses this. I know it's for guitar, but I think it's applicaple to bass.
It's called the light finger.
Basically, set all your fingers in a position (somewhere towards the middle of the neck), but don't press the frets down.
Just leave them all lightly touching the strings, with no pressure.
Then practice lifting each finger and bringing it back down, without moving the other fingers.
Other than that, try to find out what makes your fingers go nuts and slow it down while trying to keep them relaxed.
Good luck.

This is a good one and you can do it anywhere. I remember I used to do it on a table top or chair arm anywhere I was.

Thanks