I'm a long time Trekkie; I wasn't alive yet when TOS originally aired on TV, (I mean, that was the '60s -- I'm old, but I'm not that old!) but I grew up watching it in syndication. I also watched TNG, DS9, Voyager and (ahem) most of Enterprise, back when these shows first aired on broadcast television. The notion of binge watching wasn't even a thing, yet. That said... I'm somewhat in the same boat as you for some of the newer shows, simply because I haven't taken the time to sit down and consume all of "NuTrek" quite yet, so I do get where you're coming from. (The rest of "life" sometimes gets in the way of good entertainment, eh?)

But at some point you have to a acknowledge that you're fighting a two-front war. You say that you don't want to watch a series that is incomplete or at risk of an abrupt cancellation, which I certainly get... but at the same time, you say that you don't want other people who have watched it to spoil it for you, while still having the option to discuss the shows you have watched with those same people. These are largely incompatible whims; you're kind'a going to have to just choose a path and accept that there will occasionally be thorns in the bushes along that path, regardless of which one you choose.

That is to say, as I see it, you have three basic options:

• Either watch things according to your preferences at whatever pace suits you, and accept that the occasional spoiler is going to be inevitable,
• Or binge everything that's ever been released as quickly as you can, and accept that you're going to end up watching and investing in some shows that may not last as long as you'd like,
• Or simply pull out of the discussions as soon as you realize that a spoiler is coming... or maybe even isolate yourself from those discussions entirely, until such time as you're comfortable with your level of Trek knowledge.

Some combination of those options is probably going to be more or less palatable to you. (Personally, I go through spurts of all three modes at different times.)

As an aside: shows being cancelled prematurely is by no means unique to the Age of Streaming. The original Star Trek series is a prime example... but more modern examples exist as well, including non-Trek franchises, like Firefly.