Most likely needs a relay. It's a rare one, and expensive where you actually can find it. This creates an incentive to attempt "fixing" the relay.
Never makes it better, in my experience. Once the relay's spring fingers get bent from being "adjusted" the timing will never be right.
Barkett has it listed, near the end of this page:
http://stores.goldeneagleradios.com/tram-base-radio-parts/?sort=featured&page=3
One other possibility. If the neutralizing capacitor C706 has been tweaked to max the wattmeter reading, the final will become unstable.
C706 is properly adjusted to prevent the final tube from oscillating, especially in sideband with no modulation. Can create quirky symptoms in AM mode.
When it's adjusted properly, there will always be some friction on the adjustment screw. If it's been cranked so far counterclockwise that there is no friction on it, this is what happened. And if you find there is decent friction on the screw head, just return it to the position where you found it.
If you plan to use the radio for long at all, you need a spare relay anyway even if the relay in it is still okay. They eventually wear out. And if yours has a 4-digit date code stamped on it that begins with the digits "75", "76" or "77", it's probably factory original. It's rare to see that relay last more than 40 years.
Really rare.
73