I enjoy juggling a half-dozen regular bands, but subbing provides my greatest satisfaction as a musician these days, in particular when I do it on a short notice emergency basis: no advance rehearsal of course, just me doing homework. It's rewarding to receive effusive thanks for saving a band from canceling a gig because their regular bassist suddenly became unavailable. Two Saturdays ago I got the love from a bassist buddy who woke up sick as a dog and gave me his gig that evening (fortunately I was open, and had notes from subbing with his band a year ago).
Full disclosure: I'm not a jazz guy or full-time pro, I can't sight read, and I'm not great with proper chord charts (I prefer numbers to letters, see below). I mostly do bar gigs with classic cover bands who play rock, pop, blues, country, R&B, oldies. Exception: a few years ago I subbed on 2 weeks notice with an all original blues/funk/rock trio. As it turned out these sub gigs were an audition to replace their departing bassist, and fortunately I played well enough to become the primary bassist (an originals band that gets paying gigs... shocked the hell out of me

).
To the OP: sure, subbing was stressful for me too at first, but the more you do it, the more comfortable you become. You get the hang of listening, which allows you to adjust on the fly to changes in structure, anticipate endings, etc.
Another thing to remember: of course you want to play your best, but you're a sub, so don't sweat any hiccups that may occur, especially if you're coming in cold on short notice: the band probably won't expect perfection. Now, I'm saying this as a bar band guy -- the bar is set higher for big money gigs (which are above my pay grade). But once I became comfy with subbing at my level, I've been told a couple of times that I played better than the regular bassist (which might reflect on the regular bassist more than me, but I'm happy for these compliments regardless).
With cover bands there's occasionally negotiations on the set list: e.g. "I'm giving you our full list of 80 tunes, I've highlighted 20 that we really want to do, and you can pick any 25 from the remainder". But as an
old experienced vet I usually know 60-90% of a cover band's entire catalog already, and I enjoy doing homework on the remainder (depending on how much time I have to prepare) because the more songs you learn, the easier it becomes to learn new songs**.
Before my 1st sub with a band I always confirm that it'll be OK for me to read from my tablet. It's always been OK. After 8 years of using the tablet (before that, paper) it contains over 700 song "charts" -- quotes because I write them in my simplified Nashville Numbers shorthand. I need to stress: I rarely need to use the tablet for more than a few songs per set, and when I do, it's usually a quick glance as refresher. However I do sometimes have to read all the way through a chart if the song is complex, and/or brand new to me.
** I should add that writing out these charts has helped me become better at learning songs, especially when it's a challenge to parse a song into verse, pre-chorus, chorus, bridge, etc. I actually used to get annoyed if a band decided to veto a song after I'd already learned it... that was so silly! I now save a chart for every new song I learn, no matter how simple ("12 bar, quick 4"), because even if I don't play it at the sub gig it might come up again in the future.