Tapeworm doesn't usually cause issues to sheep, but what you're seeing is probably segments of dead tapeworm. The white wormer will probably have killed it. R.e. next wormer- monitor their burden with faecal egg counts and only treat if the vet tells you to when they do the egg count. Regular worming of lambs is a. often not needed and b. will speed up resistance to wormers which you definitely don't want.
Each farm is different, but for the last few years my lambs have only had a white wormer in late spring and not needed any additional treatment until well into the autumn, but as said we monitor with faecal egg counts and treat accordingly.