How dire (or not) was their situation militarily in their final 100 (or so) years of existence?

Very dire. Byzantium survived its final century more or less by luck and accident.

By the 1350s, the Empire was reduced to Thrace and a handful of isolated Balkan enclaves and islands, plus the inland city of Philadelphia in Anatolia. In the previous decade, the Ottomans had conquered the rest of Byzantium's Asian possessions. Serbia under Stefan Dusan, its most powerful medieval ruler, conquered the western Balkans but his realm rapidly fragmented after his death, creating a power vacuum that the Ottomans filled.

Byzantium in the 14th century was almost constantly wracked with damaging, distracting civil wars. Andronikos III (r. 1328-1341) was a fighter, but he died early and left the empire to his 9 year old son John V (r. 1341-1391). John V ended up fighting against his father's ally (John VI Kantakouzenos r. 1347-1354), his own son (Andronikos IV, r. 1376-1379), and his grandson (John VII, r. 1390). These were expensive and often resulted in Byzantium's neighbors taking sides for their own gain.

In 1356, the Ottomans used an earthquake as an opportunity to establish their own foothold in Europe on the Gallipoli peninsula. By the 1360s, they had conquered Thrace aside from Constantinople and Byzantium was an Ottoman vassal by 1371. This did not stop further Ottoman conquests and in 1387 they took Thessalonica and Philadelphia in 1390 (John's heir, the future Manuel II, was forced to participate in the expedition).

John V, Manuel II, and Manuel's son John VIII all spent time traveling Europe to personally beg for military assistance. This was often predicated on the submission of the Orthodox Church to the Pope, which was more unpopular than Turkish conquest to those back home.

Constantinople was besieged from 1394-1402 by Sultan Bayezid I, but he was forced to leave and confront Timur in Anatolia. Timur captured him, throwing the Ottomans into chaos for about ten years. This gave the Byzantines a slight respite, and they even regained Thessalonica for a time. Nevertheless, the city was again besieged again in 1411 and in 1422 prior to its conquest in 1453.

The Byzantines did see some success in the Peloponnese, called the Despotate of Morea. There the future Constantine XI managed to conquer the remaining Crusader States in the area and even pushed further into Athens, but the Ottomans kept him and his successors hemmed in. Once Constantine became Emperor in 1448, he moved to Constantinople and left his two remaining brothers joint rulers. They squabbled incessantly and were expelled by Mehmed II, conqueror of Constantinople, in 1460.