Is there a way to convert my new lingots (acquired after the conversion) to gems?
Yes, all my lingots were converted to gems, which I use in the app to "Master" a skill, so it won't get week for a month. But, I still earn lingots, from continuing my streak. So, now I have 166 lingots that I have no use for. I'm hoping Duolingo would let us keep converting them.
From what I have seen others say, you keep your original Lingots, but you can no longer view/spend them in iOS. However, you can view/spend them on Android/web/etc still. The 'conversion' is really about giving you some free Gems to get you started. The rate seems to be: round down to the nearest 50 and add 500.
I have watched my lingot count to see if when I earned gems that I was getting more lingots but I really haven't paid much attention going the other way.
When you finish a skill in the iOS app, you get 120 gems. When you finish a skill on the website, you get 2 lingots. If you finish a skill on the app, you earn both. I have not checked to see if my gem count has been increasing because I don't use them.