I told my old trusted doctor (Dr. Chakwas) to stay behind but didn't get prompted of gaining an asset. I then tried to recruit the younger one (Dr. Michel) and she keeps saying she has to think about it. Does she join later, or am I glitched?
Recruiting Doctor Michel to serve on the Normandy requires the use of a Charm or Intimidate dialog option (top left or bottom left of the wheel, respectively.) If you don't see either of these options in Blue (Charm/Paragon) or Red (Intimidate/Renegade), it means that Shepards reputation isn't strong enough to convince Dr. Michel to come aboard.
Go do some missions, improve your alignment, and then try again.