The short answer is that Vikings weren't superior warriors to the Saxons. If anything the "Saxon militias composed of farmers were, more often than not, successful against the Vikings.

The Danish forces in England from the 840s-870s aren't successful because they're going up against peasant militias - in fact the peasant militias wouldn't be instituted until the 870s - they're successful because their rapid manoeuvrability and flexibility gives them a strategic advantage over the professional English armies. In fact, the Vikings who attack England prior to the arrival of the Great Heathen Army in the 860s rely on being 'superior' warriors as little as possible: Viking strategy essentially involves running away as much as possible, relying on superior speed and manoeuvrability to strike undefended targets and overwhelm small garrisons before larger forces can be arrayed against them, whittling down larger forces through exhaustive counter-marching or the need to post and reinforce garrisons. Pitched battle against an organised English army could be disastrous: the Danish fleet that attacks London in 859 routs Brihtwulf of Mercia, but is slowed enough to be forced to battle by Æthelwulf of Wessex who "inflicted the greatest slaughter of a heathen army that we ever heard of until this present day". Similarly, the force that raided Winchester in 860 was successfully interdicted by Ealdormen Osric and Æthelwulf and routed there.

The institution of English peasant militias is actually a solution to the light manouvre warfare of the Danes, rather than a problem. In 871, Asser (Bishop of Sherborne and biographer of Alfred of Wessex) complained that Wessex was being 'bled white' of manpower, not by Danish victories in pitched battle, but rather through the constant need to dispatch forces to form local garrisons and try and prevent myriad raiding by peripheral forces of the Great Heathen Army trying to tie down the English and leave them vulnerable. Indeed, the West Saxon army faired reasonably well against the Danish army in 871: elements of it won a victory at Englefield; the larger army was defeated at Reading, but clearly not particularly heavy since it was then able to inflict heavy casualties on the Danes in a bloody stalemate at Ashdown. A fortnight later the Danes won a victory at Basing, but then suffered heavy losses at Meretun two months later. The strength of the wider Danish strategy can be seen in the final pitched battle of 871 at Wilton: Alfred's army was significantly outnumbered by the Danes, having had to dispatch a great part of its strength to what Asser calls "countless small expeditions" and, although it was able to rout the centre of the Danish army and acheive victory of a sort, it was unable to pursue for fear of being surrounded and overwhelmed, and instead chose to withdraw in good order.

The institution of the fyrd - the peasant militia - was one half of a West Saxon defensive reform that also created a network of burh fortresses at strategically important locations - river crossings, road junctions, fords, estuaries - specifically to deny the use of routes of manouvre to the Danes and instead allow a network of standing local garrisons to respond rapidly to threats, simultaneously curtailing the impact of Danish raiding locally, and freeing up the regular army to wage war properly. Although technically a 'peasant militia', the fyrd was no ragtag band of farmers with pointy sticks. We know from sources such as The Battle of Maldon that the Fyrd fought in disciplined shield walls led by more professional soldiery who served alongside them. The raising of the fyrd was brought about by extending the obligations of military service much further down the social hierarchy than had previously been the case, beyond thegns and Freemen/Sokemen to all free men. (Gareth Williams in Landscapes of Defence in Early Medieval Europe has an excellent summary of current consensus on the fyrd). So, for example, a King's thegn might previously have performed service in the gesith of the King, and taken with him his own gesith of thegns Ordinary and sokemen, but those thegn's Ordinary would now be expected to raise their own gesiths from amongst their tenants. In practice, this meant that the fyrd was comprised of myriad small warbands organised at the Hundred, tything or even village level which then coalesced at Shire level to stand garrison duty at burh sites. Fyrd duty could be a great opportunity for honour, glory and material gain: for examplw Æthelstan specifically honours the Malmesbury element of the Wiltshire fyrd in the West Saxon contingent of his victorious army at Brunanburh. Sources like The Battle of Maldon also suggest that the thegns, Hundredmen or Tythingmen who led each warband within a fyrd would have picked the men they knew to be the most skilled, dependable and reliable to fight alongside. Those less suited for the battlefield were instead assigned to maintain fortifications, repair roads and bridges, man watch towers and signal fires, stand a coastguard, or contribute logistically.

The Fyrd was broadly speaking very effective against the Danes, so long as the system functioned as intended. The 'famous' Danish "victory" of 878 is limited largely to Somerset after the Devonshire fyrd annihilate a Danish fleet attempting to raid the shire at the same time, and the victorious English army at Edington the following year is comprised mostly of the fyrds of Somerset, Wiltshire and West Hampshire, which also suggests that those counties remained particularly secure overall. In 885, the Kentish fyrd successfully defended Rochester until relieved by Alfred, and in 893 the South-Eastern fyrds were responsible for harassing a Danish invasion fleet landing in Essex, curtailing its ability to raid and allowing Alfred's army to overtake it at Farnham. Those same fyrds also stormed a Danish fortress at Benfleet, capturing a number of ships and loot, as well as a number of hostages that they used to prevent Hæsten from raiding Devon. A large Danelaw force which attempts to raid Mercia that same year is coralled, overtaken and largely slaughtered by a force "assembled from every borough". Similarly it's the Mercian fyrds which defeat a Danish attack at Tettenhall in 911, and specifically the Mercian fyrds of Gloucester, Hereford and Worcester which defeat a 'great naval force' in 914, which is then finished off by the fyrds of Watchet and Porlock.

To answer your broader question, we do indeed have reference to the Danjsh armies settling down to farm: in 894, the Danish army is forced out of the Wirral because Welsh raiders had 'deprived [them] both of the cattle and corn which had been ravaged." Similarly, and perhaps most crucially, in 876 the Anglo-Saxon Chronicle records that:

And that year, Healfdene shared out the land of the Northumbrians, and they proceeded to plough it and to support themselves.