I have a cedar wall that was finished with a brushed-on coat of uncut
linseed oil. When I tried to refresh a part of the wall with a second
coat, maybe 10 years after the first, the result left some shiny,
sticky patches. Now, several years later still, I'd like advice on
1) eliminating the shiny-sticky bits, and 2) refreshing a linseed
oiled surface without creating more such patches.

You can use a rag moist with appropriate solvent (turpentine or 99%
isopropyl alcohol) and gently rub the shiny spots, turning the rag
often, to remove the excess oil and even out the finish.

Technique is important. You have to develop a feel for it.