[NB: In this question, I use some of the terminology given here.]
I can hear (and produce) the difference between ö and o, and between ü and u, but whatever part of my brain is responsible for assigning phones to phonemes still assigns both phones in each of these pairs to the same phoneme1.
As a result, I typically won't be able to remember whether a particular letter in a word is a u or a ü, say, no matter how familiar I may be with the word. Therefore, I'm constantly asking myself questions like "is it schmützig or schmutzig? Kuhlschrank or Kühlschrank? ostdeutsch or östdeutsch? Franzose or Französe?"
The extreme specificity of my forgetfulness is the reason why I infer that, at some stage of language-processing, I assign both o and ö to the same phoneme, and similarly for u and ü (actually, my problem seems to be a bit worse with u and ü than with o and ö).
I can't think of a way to teach myself to assign u and ü to distinct phonemes.
This means that, just like with der, die, and das, I will just have to memorize the us and the üs, and the os and the ös. I'm looking for anything that will lighten this massive memorization task.
I find that commonsense counts for little. For example, it's Franzose yet französisch. It's grundsätzlich yet gründlich.
As I said, I'm looking for any tips that may help with remembering umlauts.
1 I don't have much problem with a and ä, so the title of this post is somewhat imprecise. Sometimes I do write e instead of ä, but I understand that this is true for native speakers as well.