You've already good answers about the second part of your question (lumpy texture), but I don't think that people thoroughly answered the first part:
I know how to tell when boxed pasta is done, but I'm not sure what the texture of fresh pasta should actually be like. Should it be similar to that of "Al dente" boxed pasta? Mine was weirdly soft and I'm not sure if it was under or over-cooked.
No, it should not be similar to "al dente" dried pasta. It's always going to be "weirdly soft", even when it's cooked correctly, if you're comparing it to properly cooked dried pasta. It has a silkier, softer texture.
Fresh and dried pasta are basically two different ingredients, sort of how fresh and sun-dried tomatoes, or fresh and dried herbs don't cook up the same way.
Usually it only needs a minute or two in rapidly boiling water (do not try to start it in cold water), with thicker or semi-dried pastas (made earlier in the day) taking a little bit longer than stuff you just rolled out.
I would recommend that you ask around, and see if any of your friends (or their parents or grandparents) make fresh pasta, and see if you can watch them, or at least sample their finished product, so you know what it's supposed to end up like.
You could also try checking to see if any restaurants near you serve fresh pasta, and try it there. (although if you're currently under a lockdown / takeout only situation, I would wait, as the texture is going to suffer). And it doesn't need to be an Italian restaurant -- there are plenty of fresh asian noodles, or even pick apart a dumpling made with fresh pasta so you can get a sense of just the wrapper texture.
And if even that's not an option, you can try looking through the refrigerated or freezer section of your grocery store to see if they have prepared fresh noodles, so you can compare that to your attempt. If you don't see any, ask -- I've seen it tucked away in the cheese case before.