Would starting my next months birth control before my period is over cause my birth control to be uneffective?

No - it would be OK. You will still have effective contraception, you will likely just have a shorter period. Some women do an extended cycle with their ocp's by discarding the placebo pills and starting a new pack - they don't have a period at all. After 3 or 4 packs of pills, they take the placebo pills and have a period.