popco - in addition to what hacksaw said, you need to sheath your thermocouple - or use a non-contact temperature measurement.
The reason for the loss of stability in readings is the electrochemical processes occurring in the nitric acid around the thermocouple: on one hand, you are trying to generate a thermopotential against the cold end; on the other, you have Cu and Fe in an acidic environment, which will inevitably cause you to lose some charges into the acid (more so from the Cu, as it is more reactive).
I could write out the exact chemistry of the processes you have, but I would need to know the concentration of the acid and the temperatures you are trying to measure.
As Cu is losing Cu+ and / or Cu++ (at a higher rate than Fe is losing Fe++ and / or Fe+++), and as the nitric acid is taking the cations, it loses some electrons off the Nitorgen, and therefore turns into an oxide of nitrogen (NO / NO2), which is a gas. That occurs on the surface of your TC wire, which is now a set of electrodes. The gas blocks the wire from contacting the bulk of the liquid nitric acid, and therefore, you are losing stability as the bubbles form and leave the surface of your thermocouple.
In addition, as copper and iron are getting etched out, they redeposit themselves as oxides / nitrites / nitrates on the surface, which degrades the sensitivity of your temperature measurement.
My best advice - once again, use a glass-sheathed thermocouple. A thin layer of glass will be sufficiently thermoconductive, and all you need is to look at the chart for this specific sheathed thermocouple.