In my own experience learning Dutch when living in The Netherlands (were, like in Denmark, almost everybody speaks good English) you learn very little and very slow with formal lessons and a lot very fast in situations were you have to manage with the local language (basically sink or swim).

I spent years living there with only basic Dutch and then ended up in a small company were I was the only non-Dutch person and the meetings were conducted in Dutch and within 1 to 2 months my Dutch language skills had taken a massive leap forward.

I also get similar effects with other languages I speak when I go visit those countries: persist in talking to the locals in the local language and that will push your language knowledge up.

That said, at the very beginning language lessons will give you the basic structure for the language, but for going beyond the basics I find that just being forced to use it yields the fastest improvements.

(Might wanna try to start watching local TV at some point too)

By the way, if the Danish are anything like the Dutch, they'll pick up from the accent that a person is American and switch to English. Do not follow them! Keep talking in Danish even if it feels like it's pretty bad and hard to use. When I lived in The Netherlands most of my British acquaintances had really poor dutch speaking skills even after over a decade there because of this effect of people picking up their accent and switching to English.