You have just slightly mistranslated several different parts of the sentence, but they've added up to make something which is a little bit off the mark. Let's look at this clause by clause.
若い人はウイルスがうつっても体の具合が変わらなかったり、病気が軽かったりする
This うつる is referring to catching the virus, not spreading it, and this whole clause is talking about what happens to a hypothetical young person if they were to catch it. The も here is perhaps better addressed as ても, which here (and typically) means even if. You can see some simple examples of its usage here. A translation of this whole clause might look something like this:
Even if young people catch the virus, there is often no change in their physical condition or they have only minor symptoms.
Finally, addressing たり, there is really no meaningful difference between its usage in adjectives or verbs. It still functions as a method of incomplete listing; the aforementioned lack of change in physical condition or minor symptoms are a subset of some possible range of phenomenon.
知らないうちにウイルスを広げていることがわかりました
This is a little bit awkward to produce a direct translation for, but I would translate it as something like this:
(We) realized that (they) had been spreading the virus without knowing it
Looking at a broader range of possible translations for 知らないうちに here may help. It's important to note that in this case it is presumably the young people in question who are the subject of this 知らない, as evidenced by the fact that they are definitely the subject of the 広げる that comes after it.
So, putting everything together:
(We) realized that because young people often exhibit no change in their physical condition or have only minor symptoms even if they catch the virus, they had been spreading the virus without knowing it.
