Just to reiterate, pps, please don't let him cry himself to sleep - no one should have to do that
I know that it's hard because you're completely shattered and suffering yourself but he really does just need a little help. Hugs though honey. I know the frustration well.
A five month old often does still need to feed during the night, I would recommend paying attention to the timings of his wakings. If your lo is waking at the exact same time, this is an indication of habitual waking. If there is variation to the timings, this is more likely down to genuine hunger.
Having said this, some babies (as do adults) will drift in and out of deep sleep throughout the night and when a lo has trouble self soothing, this can result in NWs which aren't necessarily habitual OR hunger.
At five months, he shouldn't need to feed more frequently than every four hours so that's a good rough guide for you to bear in mind.
I'd recommend (and it's how we finally battled a similar problem), paying attention to the time. If it's been four hours since his last feed, offer the breast and as tempting as it is to just give him a quick feed and put him down, let him take a full feed and offer the second breast when he's done with the first. Let him feed for as long as he will. If he wakes before the four hour mark and you know that he had a full feed previously, I'd try sh/pt in his cot. If this doesn't work, try pu/pd.
Also, don't ever rush a feed during the day. Make sure he takes as much milk as he will. This was part of my problem. Mae's always been a super efficient feeder so I was finding myself just getting on with things when she first came off. Someone mentioned about milk intake during the day and I then started offering her some more when she first came off. She took it and that has helped.
You need to be aware that sleep training can be absolutely shattering and it can be somewhat soul destroying. BUT it generally does work. You need to be consistent, and you also need to be kind to your lo and remember that he's just a baby and he doesn't know what he's doing to you
And FWIW, I know exactly where you're coming from as I've been there so much with my youngest. Hugs.
Keep us posted