Definitely, definitely, definitely crate her. Too much room is overwhelming for a puppy. Especially one with separation anxiety. The fact that she defacates and urinates immediately definitely indicates that you're not dealing with a puppy that just had to go and went, you're dealing with separation anxiety.
My boy has MAJOR SA. Read some of my previous posts for details, but he would injure himself and has to be medicated. We're using fluoxitine (prozac), which works wonderfully. He is crated when he can't be supervised because he is a threat to himself (and my sanity) if he's not.
You MUST crate train first, though. Don't just throw her in the crate and figure that will solve your problem. It won't. You need to slowly adjusting her to the crate by creating a positive environment. Do a search on crate training here and you'll find much detail on the process of crate training. Never leave her in the crate with her collar on, just in case she tries to escape, you don't want her to get strangled. SA dogs will do very harmful things to themselves to escape.
The advice BasilStreets gave you is right on. Definitely try to implement the things he/she detailed, especially making comings and goings non-eventful. Try to find The Dog Listener by Jan Fennell and implement her methods for comings and goings.
Finally, Bach Flower Essence is a good idea. You don't need a perscription. It's just flower essence. You can find it at health food stores, holistic healing shops, etc. They are for humans, but are very useful in emotional problems in dogs. I swear by Vine for aggression and dominant behaviors in dogs.
For SA - It sounds like Cherry Plum might be useful. I'm not an expert, but I found that the ones I've tried definitely work. They are benign, meaning, they can only help, they don't hurt, so you might as well do some research and give it a try. I just did a google search on "bach flower essences for dogs" and started my research there.
As always, behavior modification is the key to long term behavior change, so definitely work on those things in parallel to any medical/holistic form of treatment.