How do you handle 'stød' when your brain is already maxed out?

I'm a busy professional with only 20 minutes a day to study. I've finally reached a point where I can read basic Danish news, but the 'stød' is completely throwing me off when I try to listen to podcasts. Is it worth obsessing over the glottal stop at my stage, or should I prioritize getting the vowel reduction right so I don't sound like a complete outsider?