[nickbrickmaster]

Like I said, give your Pi a static IP address, or just replace 10.TE.AM.XX with raspberrypi.local.

That will definitely be an issue. You're right, putting bind() back in the client will fix it. Here's my code that I tested this with. Uses python 3, so might be a little different than yours.

Client

import socket

s = socket.socket(socket.AF_INET, socket.SOCK_STREAM)
s.bind(('', 5800))  # Bind to empty IP so it can use anything
s.connect(("192.168.1.3", 5800))  # 192.168.1.3 is the IP of my server
s.send(b"hello")
s.close()

Server

import socket

serversocket = socket.socket(socket.AF_INET, socket.SOCK_STREAM)
serversocket.bind(('', 5800))  # Again, bind to any address, but this specific port
# become a server socket
serversocket.listen(1)

(client, addr) = serversocket.accept()
print("Recieved connection from {}".format(addr))
while True:
    data = client.recv(1024)
    if len(data) == 0:  # According to the spec, empty recieve means that the socket is closing
        break
    print("recv {}".format(data))
client.close()

According to Wireshark, this code only communicates on port 5800.