Before publication, if a new edition of a source is released, should I try and find the new edition or keep a citation based on the old?
You should cite the exact source of any information you use. If you are pulling your information from a 200-year-old Encyclopedia Britannica, you cite the 200-year-old Encyclopedia Britannica.
However, it is good practice to write based on the current state of knowledge. If the 2010 Encyclopedia Britannica has more up-to-date information, you should use that more recent information, which will drive you to cite the more recent edition. That doesn't mean you need to wait for a new Britannica to publish, but if a new one does appear before you go to print, you may want to spot check it for relevant updates.
(In most fields you probably shouldn't be basing any writing off an encyclopedia, but it's an easy example to think about).
I mainly agree with the first answer by SolarMike, but be aware of the content/information that you cite in this old edition and how old your edition is. The state of the art might change due to new developments in your field and this will be included in the edition. If you, for example, then criticise the book for not including X or Y, but it is done in the most recent edition, this would have been an easy mistake to avoid.
Regarding the year of the edition, sometimes, valuable editings are made for the newest edition of a book. It might be interesting for you to check these - just that you are aware of. For instance, the author changes this specific argument that you use due to criticisms of the first edition (or puts in an extended chapter or so).
There are of course cases where you might want to deliberately cite an older edition. For example, if it contained information that has been removed from newer editions; or if you want cite erroneous information "Until recently it was widely believed that.... This belief appears to derive from ....". Or you might want to cite Fowler 1st ed as evidence of what was considered acceptable English usage at the time it was published. Or you might believe that earlier editors got something right and later editors got it wrong.
I can also imagine cases where you want to cite an earlier edition simply because it is much more widely available, or because you cannot get hold of a later edition despite best efforts.