The safest quick way to dry them is to chase your ring, cut them to rough bow size but leave the tips and handle full width. For mulberry I'd go with 1-3/4" wide. Then get the limbs 5/8" thick for the entire width and length of the limb. NO THICK SPOTS and NO SAPWOOD allowed. Seal them with shellac on the backs and ends. Leave them in coolish spot for 1 month and then slowly move them to progressively warmer and dryer places. Like start in a basement for a month, move to a unheated garage for a week, move them into a closet inside a heated home but one that doesn't have an air duct for a week, then in a room with some airflow for a week, and finally to the warmest place in your house.
The absolutelely quickest way to dry them is cut to final dimesions, get them less than 5/8" thick, floor tiller green and move straight to the warmest place in your house. Watch for checking the first couple days. They'll be dry (ish) in 2 weeks. You won't produce the best bow with this wood but you will be making bows.