I had a similar problem when I got the 973A off ebay when it came with old long time dead batteries installed and the usual blue-green corrosion on the battery clips. I actually had to remove the battery holder by unsoldering the connecting tabs from the main PCB (remove LCD display first), as it was really the only way to be able to completely remove the whole mess which in the end cleaned up quite well after using some diluted acetic acid (common vinegar), an x-acto knife as a scraper and a small brush. After that the DMM would seem to function but the lower uA and mA current ranges where not displaying zero without any input, instead displaying some value, but everything else seemed to work. It appeared that the outgassing of the old batteries had been spread around leaving a seemingly invisible layer of conductive and corrosive electrolyte in some areas and was bypassing or creating some unwanted electrical paths.
After some careful inspection also fund a couple of very thin copper tracks had been partially eaten away and still had tiny blue spots at random points on them, located somewhat near the top of the board near the display, and also between the display and the rotary selector switch area. I had to use a 10x magnifier in order to get rid of all the really almost microscopic tiny green spots here and there along those tracks as they where so small that they where virtually invisible otherwise. I first cleaned all those areas with Qtips and alcohol, but some of the tiny corrosion spots really required some gently scraping off with the sharp tip of a x-acto knife, and then proceeded to patch up one or two spots on the traces which looked suspect.
Unfortunately after all that work the current ranges where still not zeroing, but at least I had moved forward in cleaning and fixing those traces. Next step I took out the whole main board, removed the display and battery holder, and carefully washed the whole board down with a solution of water and dish wash liquid (Palmolive) with a soft tooth brush and gently going in circles letting the soap do its thing. Finally rinsed off everything under running water and thoroughly let it dry off. After reassembling the unit, finally this time the current ranges started to zero properly, but then I noticed that some of the other volt and ohm functions where a bit off calibration. So did a full calibration on all functions and ranges with the help of the trimmer adjustment locator in the manual, and ever since the DMM has been performing well so far.
One of my 974A also came with old corroded batteries inside, but for this one I only had to remove/unsolder the battery holder to remove the blue stuff off the contacts, and do some general cleanup with alcohol/Qtip around the battery area for good measure, and that was enough to make everything work. BTW I now do have total four 974A units after not being able to resist buying a brand new one some time ago at a very reasonable price on ebay still in the box with original test leads, and which I have set aside as a future "spare" unit. Unfortunately nowadays all the asking prices seem to have gone through the roof ($100-150+) even for a no-leads, as-is 973A. This in comparison of what they where selling for perhaps 4-6 years ago rarely going over $50-$75. I remember actually paying just $99.99 for my first used mint 974A back in 2014, including the original box, test leads, and user manual. Now in more recent times, probably the resellers have been getting wise to the real quality of these DMMs, and as usual are trying to further milk the cow.
After having some bad experiences with supposedly good quality brand name alkaline batteries like Duracell and Energizer, I no longer trust any of them. They are either not made as they used to, or it would appear that the chinese have managed to flood the global markets with counterfeit batteries that look exactly the same as the originals so one never knows what you are really getting. Anyway, to be on the safe side, and also to cut down on recurring battery cost, for some time I only use AA or AAA NiMH rechargeable cells from well known brands like Panasonic/Eneloop on all my important portable electronics, also including all my HP Led calculators and the HP41CX N-Cells which I nearly also lost to some battery corrosion that formed only weeks after replacing the batteries with some "fresh" ones. The HP973/974 DMM's really don't seem to care about the difference of 1.25V against the regular 1.5V per cell, and even if there is some difference in running time, they usually still last for months before needing a recharge, so really who cares.
I am always amazed of how accurate these old DMMs still are, even after all these years since they left the factory in Japan. Specially the 974s, they all are almost dead on compared to my HP3468A/78A 5.5 digit MMs. Too bad the schematics where never released, probably forgotten in a drawer somewhere in Japan under the Yokogawa brand name.