Jag har en dryck literally just means that you have it in your possession. English is a little bit odd in that we often say "to have a drink" to mean more or less just "to drink" or "to have a look" to mean more or less "to look". This is not what this sentence means in Swedish. Att ha en dryck is just talking about the simple possession of a beverage, not the act of drinking anything.
Yes, spoken Swedish sounds like that. Just keep putting some time and effort into it and you brain will automatically start to adjust and learn. Part of learning a language is about that, learning to separate the words in spoken form. It's a bit of a threshold to get over, but once you get over it, it's very rewarding!