I had the same issue when I started. What I used to do if I was having trouble keeping up in the lesson is use my fingers as your doing, but then try the pick after. Get the thinnest you can find and it will be easier on the upstrokes and slow your tempo down so you don't get hung up as easy. You really should get used to a pick, although Anders teaches a finger strumming technique in Acoustic Level 1 but that's later on for you.

Moe

P.S. I once read a story that Billy Gibbons of ZZ Top used a Mexican Peso cut in half as a pick. That's what gave him that unique grungy sound. Don't know if it's true, but I could see him doing that.