As with others I will share my experience, though nothing says yours will be the same, of course.
A little over a year ago we introduced a new cat into the house. I'd had mine here for 14 years, my new husband brought his 11 year old with him when we got married. We'd tried to introduce them gradually, sharing blankets back and forth betwee houses before he moved in, bringing his cat here in her carrier and just having her in the house, contained, a few times. Still, nothing prepared us for the uproar we had. There was ugliness and chasing and hissing, and growling. My cat was the one making all the noise, his cat did the chasing. Both had been the ruler of the roost previously. I felt so badly for my cat, who is the more anxious one, and seemed to be being shoved into a corner of her home all the time. She'd hide. She didn't eat. She didn't groom herself. It tore me apart!
We did try, eventually, to isolate my cat to her "safe zone" of the room she was always running to when chased. We put a litter pan in there, but only AFTER she'd started going in there w/o the pan. We added food and water, and a cozy spot for her to curl up. She used them sparingly.
At the vet's suggestion we purchased Feliway Spray and that did help to some degree. Things have gradually gotten better, after a year and a half. There is still growling on "my" cat's part, but no longer do we have the chasing. They can be within a couple feet of each other and just kind of move around each other till one moves on to her next spot. At this point it's a comfortable situation, and I'm happy they have come this far together, even if they never become friends.
I wish you the best of luck! It's just an awful feeling to have your sweet friend acting scared and acting out, even if you understand the reasons why!!
We added food and water, and a cozy spot for her to curl up. She used them sparingly. 
