The dinghy comes to a halt and the sun regains its intensity. It beats down as I squeak along the side of the boat – the smooth rubber of my wetsuit grating on the coarseness of the bow. We're 22 miles from port – out in the open Atlantic – but it feels claustrophobic.

I watch as the skipper pulls out a bait box filled with an assortment of fish guts and gills. He flings it from the side of the boat and dunks it up and down as blood floods out of the holes and into the blue. Then he tears open a plastic bag of bait fish and begins ripping into them, squelching out the innards and tossing the mush overboard. The stench is salty, putrid.

"Now what?" I ask.

"We wait."

And there we sit – bobbing – in silence but for the plop of the current against the boat and the occasional slosh of fish blood thrown overboard. As we wait I go through the drill again in my head: "Descend on the line, stay upright to make yourself look bigger, tuck your hands in, stay away from the chum and if one gets too close, slap it in the gills."

I can't imagine ever being brave enough to slap a shark in the gills but it's this piece of advice that sticks in my head the most.

Perhaps it was idiotic to do this. I am utterly, pathetically terrified and have already enjoyed three days of diving off the coast through the volcanic rock formations that surround the Azores. I'd been told this complex of nine islands offered some of Europe's best diving, and there's been no reason to disagree. I've swum through giant crevices, spotted eels hidden in the cracks, watched juvenile eagle rays glide by overhead and sat on a boulder 30m below surface, watching a green turtle just below me. This is one of the few locations in the world where, in July, August and September at least, diving with oceanic blue sharks can be almost guaranteed.

Enough contemplation. There's no backing down now, anyway.

The boat is half empty, in a deliberate effort to keep the wildlife calm. A German couple whose names I've already forgotten sit opposite, smiling calmly – probably sharing a joke at my obviously worried expression.

Then the shout goes up: "Shark!"

And from the side of the boat the silhouette is unmistakable: it swerves towards us, dorsal fin emerging from the blue, swipes at the bait fish and circles again.

"Grande," cries the divemaster.

"Shit," I think.

Time to get in.

The Germans slip on their gear and drop off the boat in minutes. I stall and crash into the water – it's far from the calm entry I'd been instructed to make.

"Get to the line," shouts the divemaster. But I've already taken a first sight of the shark through my mask. It's loitering about half a metre from me (the Germans tell me later that I practically landed on it as I jumped in). I'm transfixed, and have to be all but yanked over to the line, where we then drop down to 10m.

You could be forgiven for not knowing the Azores exist. The nine-island complex protrudes from the Atlantic midway between America and Portugal, to which the islands belong. They are mostly associated in Britain with winter sun and pensioner tourism, offering (as they do) warm weather most of the year. Before I leave I am told to expect a Saga experience, and the half-full weekly flight from Gatwick is populated almost entirely by silver-haired holidaymakers.

But while plunging into shark-infested waters is the most extreme thing I've done here, I'm engrossed by all aspects of the Azores.

I've toured the region's main island of São Miguel, visiting an old tea factory high up in the mountains, which still functions, using antiquated British belt-driven machines. I've learned the brutal history of the Azores' traditional whaling industry, and stayed in a converted 17th-century monastery.

The undulating volcanic landscape and carpet-flat ocean make nearly everywhere a photo opportunity. We've driven along roads lined with pastel-blue hydrangeas (native to China), ginger lilies (native to central Asia), Pohutukawa trees (native to New Zealand) – all growing wild – and passed milking stations where Jersey cows graze on stepped fields. It's the sort of setting that wouldn't feel out of place in a decent Wes Anderson film.

But the islands are in crisis. Unemployment is high and locals working in tourism told me the already fragile industry is beginning to dip. By the imperious marina in São Miguel stands a partially constructed super-hotel and shopping centre surrounded in scaffolding and abandoned, without anyone seeming to know why.

Further into town, fresh graffiti from the right-wing independence movement the Azores Liberation Front adorns a number of buildings – the last time the group generated any interest was back in the 1970s.

Pushing the adventure tourism market – of which diving with sharks is key – feels like part of an attempt to revive the island's hopes. There are new hiking and cycling holidays, too.

As I descend down the line, still frantic with panic, I realise there is not just one shark circling below, but five.

I cling to the rope; suspended above open ocean half a kilometre deep, I'm a dot in the blue expanse. One shark heads straight towards me. She stares me out, her massive pupils within a foot of my eyes. I catch a glimpse of her jaw and watch her glide around me, then look up to see the sun splintering down on two more of them.

It's at this point, paradoxically, that the fear begins to subside. Beneath the waves you can see how wonderful these creatures are, perfectly streamlined and elegant. I look back to see a three-metre female (the largest my divemaster has ever seen) attacking the bait box that hangs in the distance. This goes on for an hour, a lightly choreographed dance where the sharks come so close I really could have slapped one in the gills, then circle around me and head to the bait. Eventually they return to the deep and we to the boat. I am speechless.

After the hour-long ride, back at shore we are greeted by the dive shop owner Norberto Serpa. He strides over to the quay accompanied by his two labradors; his long, matted hair is wrapped in a bandana and bounces along as he goes. He's eager to hear all about it but I'm coy with the details – still embarrassed by how scared I'd been. He sits me down, plies me with gin and tells me about all the wildlife I haven't yet seen: the giant manta rays commonly spotted in September, and sperm whales giving birth off the shore in spring. (Serpa works as a research assistant at the University of the Azores.)

So how does he feel about the islands becoming push to market these islands as a destination for adventure tourism? Surely it'll ruin the wonderful marine experience?

"I like things just the way they are," he says, "I don't see the need for more to come." I'm with him on this, but I wonder how long it will be before these unique tales of close encounters in the ocean become more of a commodity among visitors.

• Original Diving (020-7978 0505, originaldiving.com) provided the trip. A seven-night holiday on a B&B basis (two sharing), with flights from Gatwick and car hire, is from £650pp