The word "senji" (https://jisho.org/search/senji), for example, has more than 20 different definitions with realtively few similarities between them. I understand that words can have multiple meanings, in any language, but I'm confused by words that have this many with such variation in their meanings. Can I safely pick any one of these meanings and make a case for it that it means that (specific one) in the specific context that I'm using it in? Or are most of these meanings not really that accurate?
I suspect you haven't got round to learning the Japanese characters yet. In the page you are looking at there are only four different meanings for the word 'senji'. The other sixteen are different words. Furthermore, if you look at the kanji 戦時, 潜時, 選時, 宣旨 you'll see that they are four words with completely different spellings.
That being said Japanese does have a lot of homophones and the primary way to distinguish between them is context as you noted.