In applying a 3/8 sleeve anchor wood to cement.
Should you use the masonry 3/8 drill bit to drill the wood too?
Or switch over to a 3/8 wood drill?

Use a regular fluted bit for wood, masonry bit for the concrete. You may find it tough going with a regular drill in concrete, impact drill for concrete is better. Even better is an SDS shank drill and bit. One problem with a regular masonry bit is that it may hit a stone, shift a bit and then it wallows out the hole and the sleeve doesn't grab the side and will pull out.

Thanks I also have a hammer drill for cement. I was thinking about using Phillip Tapcon screws. The anchors must be flush with the wood, because a metal piece fits over the wood strip for a screen door frame.(Could countersink) the wood has pulled out of the cement that had plastic anchors. With this loose shifting strip of wood, has caused the door not to close. The filler was used,because the door width was too short.