Eärendur, son of Elendur, was the last king of Arnor per se, dying in 861 of the Third Age according to Appendix B of The Lord of the Rings. The Appendix further notes:
After Eärendur, owing to dissensions among his sons their realm was divided into three: Arthedain, Rhudaur, and Cardolan. Arthedain was in the North-west and included the land between Brandywine and Lune, and also the land north of the Great Road as far as the Weather Hills. Rhudaur was in the North-east and lay between the Ettenmoors, the Weather Hills, and the Misty Mountains, but included also the Angle between the Hoarwell and the Loudwater. Cardolan was in the South, its bounds being the Brandywine, the Greyflood, and the Great Road.
Thus, by 1975 Arnor was actually in three pieces. The three were conquered, or depopulated, at different times; the Appendix tells us:
A great host came out of Angmar in 1409, and crossing the river entered Cardolan and surrounded Weathertop. The Dúnedain were defeated and Arveleg was slain. The Tower of Amon Sûl was burned and razed; but the palantír was saved and carried back in retreat to Fornost, Rhudaur was occupied by evil Men subject to Angmar, and the Dúnedain that remained there were slain or fled west. Cardolan was ravaged. Araphor son of Arveleg was not yet full-grown, but he was valiant, and with aid from Círdan he repelled the enemy from Fornost and the North Downs. A remnant of the faithful among the Dúnedain of Cardolan also held out in Tyrn Gorthad (the Barrowdowns), or took refuge in the Forest behind.
Thus, by 1409 Rhudaur (a sizable chunk of Arnor) no longer existed as such. Cardolan was tottering, and barely existed. Next:
In the days of Argeleb II [the mid-1600s Third Age] the plague came into Eriador from the Southeast, and most of the people of Cardolan perished, especially in Minhiriath. The Hobbits and all other peoples suffered greatly, but the plague lessened as it passed northwards, and the northern parts of Arthedain were little affected. It was at this time that an end came of the Dúnedain of Cardolan, and evil spirits out of Angmar and Rhudaur entered into the deserted mounds and dwelt there.
So, by 1660 or so, Cardolan effectively didn't exist as a country any more, and was essentially deserted.
The battle in 1975, held to be the end of the North-kingdom, was thus just the end of Arthedain; the other two-thirds of Arnor had been more or less completely depopulated centuries earlier.
Arthedain itself, of course, was where the Shire and the Bree-land were; and we're given hints that there were other hobbits and perhaps other humans in the area—but as far as the rest of Arnor, there seems to have been no one else there; certainly no settlements. Orcs did become a problem, starting a couple hundred years before Bilbo's journey; but only rarely outside the easternmost areas:
When the kingdom [that is, Arthedain] ended the Dúnedain passed into the shadows and became a secret and wandering people, and their deeds and labours were seldom sung or recorded. Little now is remembered of them since Elrond departed. Although even before the Watchful Peace ended evil things again began to attack Eriador or to invade it secretly, the Chieftains for the most part lived out their long lives. ... later, in the days of Arassuil, Orcs, multiplying again in the Misty Mountains, begin to ravage the lands, and the Dúnedain and the sons of Elrond fought with them. It was at this time that a large band came so far west as to enter the Shire, and were driven off by Bandobras Took.
(All quotes taken from Appendix B of The Lord of the Rings.)