I find that the majority of my wins occur when the other team loses a member, and the majority of my losses occur when some random lvl 30 gets matched with me and carries the other team.

What I mean to say is, most of my wins have been pure luck, and most of my losses we get steamrolled by a carry. This results in me gaining levels quite quickly and getting matched with much harder opponents, without having many good, full 5v5 games to get better in.

I honestly feel like my summoner level is going up way too fast. Is this normal? What should I do? Should I keep playing on this account or just make a new one so I can be a lower level?

When you first start, you Elo score will tend to fluctuate quite a bit, leading to fewer tight games and the occasional wild mismatch. It should start to settle down before too long.

Try to play carries. Try to raise your elo more, as you get higher up in elo, leavers become less common. Don't be discouraged either, if level 30s keep beating you your elo will eventually drop enough so that you will play people who are just like you in terms of skill and who may have leveled too fast just like you.