Preventing older dog teaching new puppy separation anxiety and anxious behaviours

Hey everyone. We have a rescue greyhound (7F) who has separation anxiety from my partner, her owner. I am hoping to get a chihuahua puppy and would like to hear advice on how to prevent puppy from learning separation anxiety when they are alone together. Although she is on two medications for her anxiety, Our greyhound has a history of howling, chewing things up, and generally being quite distressed. we have worked hard to improve her distress and it is much better than it was years ago, but after losing our other greyhound (my dog- RIP Maggie💔) six months ago, her attachment to my partner has increased a bit, and despite her medications, delicious treats and activities, calming music ect, she will flop down at the front door, cry, pace and howl when we are gone (which we try not to let happen to often)

I plan on crate training the puppy, as I hear this could help. Could I also have them separated in different rooms when they do have to be alone in the house together? Increase greyhounds anxiety meds? Any advice would be greatly appreciated – and of course, if the advice is to not get a dog , I understand that too. I am hoping there is a way I can have my dream dog and make it work