Siamese have a very unique character. I have 2 myself and 3 others being persian and domestics. My first lilac point was also from a breeder who dumped him and his siblings off at a shelter. He aswell was not socialized well with humans and other cats. I started off with just him for a few months before I introduced other cats. This provided him with security and we were able to bond gradually. I then moved in a bonded pair of cats but I introduced all three very slowly by keeping the 2 in a room and let my siamese smell them through the door. After about 1 month I moved one cat out (the female) followed by the male a few days later. This introduction went relatively well with no fights but alot of hissing from him. Now they are very bonded. Later (about 1 and 2 years later) I brought in more cats again doing the same exercise.
Introducing cats is the same concept but more lengthy then canine introduction. Since your cat is stressed with these new beings, I would reconsider taking in any more until the siamese builds significant bonds with the cats you currently have. Not all cats will get along, but they will learn to tolerate one another.
Does your siamese have a preference with any of your current cats? If so, I would take the 2 or 3 and give them their own room for the time being..a safe place for your siamese to relax and not be so stressed. Give all the cats time to find their companions and space where they feel comfortable and not threatened. Introductions should be very gradual.
There are many cat savvy people here that will assist with maybe better ideas. Good luck.