Not that much is known about actual Norse rituals, compared to the mythology. The written accounts are from much later and were written by people who had no interest in documenting pagan rituals and quite possibly didn't know that much about it anyway. There are quite basic facts that are unknown and quite debated, like whether a priesthood existed. In the sense that religious rituals were handled exclusively by specialists.

That said, it is completely clear that sacrifices were an important, central even, part of the cult. Food sacrifices, weapon sacrifices and animal sacrifices were very common, and there was a ranking of animals too, with horses at the top. But human sacrifices occurred as well, although less commonly. We have lots and lots of archaeological evidence of animal sacrifices, most commonly in association with funerals. A small number (half dozen or so) human sacrifices have been found as well.

Both animal and human sacrifices are repeatedly attested both in contemporary accounts (e.g. Adam of Bremen) and also the later Sagas and whatnot. In the case of the Swedish king you mention, it's Domaldi in Ynglinga Saga who sacrifices himself at (Old) Uppsala to appease the gods after years of failed harvests, due to him having been cursed by his stepmother.

Now, this story is a pretty clear example of what's called sacral kingship, where the king often has divine origin (which the Yngling Kings do in Ynglinga Saga) and are attributed supernatural powers and the ability to bring prosperity.

Problem with this is, as far as modern source-critical scholarship is concerned, neither Domaldi or any of the other Yngling kings are historical and the prose details of Ynglinga Saga are not reliable, having been composed in the 1200s. (although based of the poem Ynglingatal which does date from the Viking Age) Folke Ström wrote a treatise (Diser, nornor, valkyrjor: fruktbarhetskult och sakralt kungadöme i Norden, 1954) trying to debunk the concept, and the existence of sacral kingship is at best a controversial topic today.

That said, there is still broad consensus that the kings played important religious role in the holding of sacrifices. This is a repeating theme in sagas, but there's also contemporary evidence such as Anund Gårdske, a christian king of Sweden who - according to Adam of Bremen was deposed by pagans in Uppsala after he refused to perform the sacrifices there. A similar thing is said in later Icelandic account bout king Inge the Elder, who was replaced with his brother, who'd promised to perform sacrifices (blót) and got the nickname Blót-Sven for it. But Inge returned with christian troops and retook power. (Obviously there's some suspicion these two accounts may be based off the same figure)

The king was not responsible for more minor everyday sacrifices though.

It is another recurring theme that the Norse peoples relation to the gods was quite transactional. You had to sacrifice to receive. The traveler Ibn Fadlan, wrote a famous account of Rusiyya (literally Rus' but likely referring to Vikings/Varangians here) in the east. Ibn Fadlan was an eyewitness to a gruesome human sacrifice in connection with a funeral, but he also wrote about more a everyday sacrifice and petition to the gods:

The moment their boats reach this dock every one of them disembarks, carrying bread, meat, onions, milk and alcohol, and goes to a tall piece of wood set up <in the ground>. This piece of wood has a face like the face of a man and is surrounded by small figurines behind which are long pieces of wood set up in the ground. <When> he reaches the large figure, he prostrates himself before it and says, "Lord, I have come from a distant land, bringing so many slave-girls <priced at> such and such per head and so many sables <priced at> such and such per pelt." He continues until he has mentioned all of the merchandise he has brought with him, then says, "And I have brought this offering," leaving what he has brought with him in front of the piece of wood, saying, "I wish you to provide me with a merchant who has many dinars and dirhams and who will buy from me whatever I want <to sell> without haggling over the price I fix." Then he departs. If he has difficulty in selling <his goods> and he has to remain too many days, he returns with a second and third offering. If his wishes prove to be impossible he brings an offering to every single one of those figurines and seeks its intercession, saying, "These are the wives, daughters and sons of our Lord." He goes up to each figurine in turn and questions it, begging its intercession and grovelling before it. Sometimes business is good and he makes a quick sell, at which point he will say, "My Lord has satisfied my request, so I am required to recompense him." He procures a number of sheep or cows and slaughters them, donating a portion of the meat to charity and taking the rest and casting it before the large piece of wood and the small ones around it. He ties the heads of the cows or the sheep to that piece of wood set up in the ground. At night, the dogs come and eat it all, but the man who has done all this will say, "My Lord is pleased with me and has eaten my offering." (Montgomery's translation)

Although Ibn Fadlan's disdain is scarcely hidden, there is not that much reason to think he misrepresented this depiction of the cult as quite transnational in nature, as it's a recurring theme. It's stated most directly in the (pre-Christian) poem Hávamál, where some well known lines read:

Veistu, hvé biðja skal?

Veistu, hvé blóta skal?

Veistu, hvé senda skal?

Veistu, hvé sóa skal?

So these lines are asking "Do you know how you shall ___" (although expressed more elegantly) where the blank are different verbs, forming alliterating pairs (Norse poetry made frequent use of alliteration, not rhyming). "Biðja" means "to ask, request", "blóta" means "to sacrifice" - so asking from the gods and sacrificing to the gods are paired here. A sentiment that seems to be repeated in the next pair "senda/sóa", where the former is "to send" - which has generally been interpreted as meaning to send a request to the gods, while "sóa" is a rare verb mainly used in ritual contexts, meaning "to waste, squander". To make a sacrifice.

The connection is made even more explicit in the lines immediately following:

"It is better to not ask / than to over-sacrifice / A gift requires an exchange / It is better to not send / than to over-squander"

So it's clear that what they're saying here is that if you want something, you must sacrifice. But don't go bothering the gods for every little thing.