I suspect that the turn signal switch pins are not in the correct location. Apparently, there is power on the orange/black wire feeding the K7 relay all the time. Verify that the turn signal switch is working correctly using an ohmmeter with the switch unplugged. Also visually verify that turn switch pins are inserted into the correct location on the both sides of the connector. A turn signal switch diagnosis guide and a 1967 Cougar turn signal switch connector diagram is available from my website: www.cougarsunlimited.net
If the brake light switch is getting warm, current must be passing through it. Use a voltmeter to check for power on both sides of the switch. If the switch is stuck on, remove it to see why.