Good morning, Toby & welcome to the site!
What you describe is actually a common issue with a cemented cast-iron stove.
Unlike steel which is welded, cast-iron stoves are assembled in pieces. Today, most cast-iron stoves are gasketed where older cast-iron stoves were assembled with cement in the seams to create an airitght firebox once cured.
What happens is over time (about 10 years or even more) the cement gives out and crumbles away. This causes the tiny seams to open up and allow air to be sucked into the firebox as the oxygen is consumed by the fire.
It is recommended that a cemented stove be torn down and reassembled every 10 years to keep the stove working like new and airtight.
Our shop does this work and can replace any worn parts you may need.
A typical rebuild can run from $400 and up if parts are required. If you're local come by and see us. If you're not, you can ship the stove to us and we can ship it back once rebuilt.
Weigh your options. With a stove that old, you will deffinitely discover many advantages to applying the rebuild investment into a newer, EPA certified model with much higher efficiency and burn-times.