DeOxIt D5 and work the knob would probably do the trick to get the heebie-jeebies out of channel switching. But the "purists" among us would probably insist on a more in-depth approach, as spraying tuner lube willy-nilly can potentially screw up any mica-compression trimmer caps if any are used instead of the cylindrical type comprised of a ceramic sleeve, and that detuning is a danger. I've never encountered such problems with "spray and twist" technique meself. One needs to use good judgement, and not empty the entire aerosol can into the thing, of course....sparing application should suffice with D5
The last Admiral set I had done any major work on (HL3) used a "standard coil" tuner, where the coils for each channel rotate on a drum, placing one coil against a battery of contacts up by the tube sockets and their associated components. Generally the vigorous wiping action associated with channel changing keeps the contacts free of tarnish. But most tuners of that design have a bottom cover that is easily removed, exposing the coil drum, where thoughtful application of a pencil eraser on the contact knobs will go far in restoring stability. IIRC In some of the tuner models the entire shaft and drum assembly can be removed after some wire retainers are loosened. In this case the stationary contacts can also receive the same attention.
How far you want to go with this depends largely on your degree of ambition and your assessment of your skills.
But while I have gone that deep into a tuner many times in early days at this point I'd probably say "Shucks, life's too short for this BS." and grab the D5 and give 'er a little squirt and a few twists.
Probably all it really needs. With the tuners that have rotary switches and stationary coils the squirt-and-twust method is probably your best shot, as diddling around in one of those with all the delicate free-form coils hanging out in there can open up a very unpleasant can of worms due to detuning.
As far as doing any work involving R&R of the tuner you need to be sure that all lead dress is exactly as you found it when you started. The IF signal is usually drawn off through an RCA connection, but filament, B+ and AGC lines usually solder onto feedthrough capacitors embedded in the tuner chassis. Care must be used in desoldering/resoldering in that excessive heat or force are not suffered by these components. Inside the tuner use extreme care that any free-form coils or any leads are not in anyway repositioned or reshaped, as such are critical in the tuning of the circuitry involved.
Many TV men in the old days never went inside a tuner to any great depth, but just removed it and sent it to one of numerous agencies that would exchange it for a reman to stick in its place. People here may chime in on this and perhaps differ--and I hope they do, but my gut feeling on this is that such services may no longer be available for that type of tuner.
Hopefully others will chime in and offer light from different angles on this, but the above observations have always worked well for me.