Do not worry, this is a very common problem on tuba. The low range of the tuba takes a long time to develop and will naturally sound better over time - given that you play regularly.
Something I always tell students, is that you have to balance your range. If you want to increase your low range, you need to increase your high range as well - you can't do one without the other.
If your high range sounds thin, you need more air and you need to relax your embouchure. Chances are you are trying to hard to force the notes out through tight lips. You're lips need to be able to resonate freely in order to produce a beautiful sound.
Regarding the low range, it is good that you do not puff out cheeks. You're correct, you need a very relaxed embouchure for the notes to speak well. Remember, low notes have a much slower frequency than high ones. Your lips need to be able to move much more slowly as well. If they are strained or tight, they cannot vibrate as they need to. When you play low, you should pretend there is a tennis ball in your mouth - really picture what it would feel like to have a tennis ball in your mouth (your jaw should drop about 1.5 - 2", your tongue should move down, and your soft pallet should move back - in a similar fashion that you may adopt at the dentist.) If you do that convincingly, your mouth and jaw should do the rest.
Three other points:
1.) Remember that when playing tuba, it takes twice as much air to play the same note at the same volume an octave down. So you need 2x the air to play A3 as you do A4. This is also true for the lowest ranges of the instrument. You should be using large volumes of air to create large, slow-moving columns that can resonate slowly.
2.) If you want to play high/low, you actually have to play high/low. Play all your études and scales throughout the entire range of your instrument. Every day. Read my answer here for some ideas that you can incorporate into your daily routine.
3.) Take a look at this video showing a part of an Arnold Jacobs masterclass. If you don't know who he is, look him up, learn from his videos, and use it.
Link to video