It varies. While in (most) Romance languages nominal morphology underwent extreme simplification, verbal morphology did not. The passive voice disappeared completely and a number of participles were dropped. Other than this, most Romance languages have a very simialr amount of verbal forms to Latin.
The synthetic future tense disappeared and was replaced by an analytic construction, which has since fused and turned into a new synthetic tense, which in French is already losing toward a new analytic construction: amabo -> amare habeo -> j'aimerai -> je vais aimer. In French the use of pronouns became mandatory as most inflections fell together, and the synthetic past tense has been relegated to literary registers.
You can look at other language families - Turkic languages are a good example of a language family that in spite of extensive contacts with ohter peoples *Arabs, Iranians, Greeks, Armenians, Chinese etc). remained very conservative gramatically
Turkic languages are agglutinative rather than fusional. This means that even when sound shifts cause affixes to erode, they get restored by analogy. On top of that, Turkic languages do not have the strong stress accent of Germanic languages or even Latin.
I think an important reason why those langauges changed so much, apart from the Black death, was that they had a lot of contact with Low German. This is especially true for the larger towns/cities. The language was simplified in the process of communicating with the Germans. Sure, some dialects changed less than others. I am no expert when it comes to dialects, but as far as I understand, at least some Norwegian dialects have the dative case and in Sweden you have some varieties of Scandinavian that must be considered a separate Language, as they are mutually intelligble with neither Swedish nor Norwegian/Danish. Still, if the language should be preserved in a way similar to Icelandic, you would need an early POD, or alternatively the elite would have to learn the language from scratch. Norwegian Landsmaal was more conservative than Swedish, Danish and the Riksmaal variant of Norwegian, but it still was far from Old Norse.
It wasn't just Low German, but also interdialectical borrowing. When you have several innovative urban centers in a Germanic language, what happens is that each center reduces unstressed vowels in a different way, merging different sets of affixes. As a result, the only way people from different cities can understand each other is to ignore these affixes and invent new analytic ways of showing syntactic role. This didn't happen in Icelandic because Iceland did not have cities.
Contact with Low German may have accelerated this trend, in the same sense that contact with Old Norse may have accelerated this trend in late Old English, but it didn't by itself cause it. After all, there was extensive contact between various Italian varieties (and between Italian and French), and yet this process did not happen in Italian after the initial transition from Vulgar Latin.
The Black Death... eh. It affected practically all of Europe and the Middle East. Scandinavia isn't special.