Both forms are perfectly fine, except fix "become" to "became" in the second one, as Brad already mentioned. In the fist example, "would suffer" is the past of "will suffer". It's common enough in documentaries and biographies, where they jump backwards and forwards in the past, relative to some event. But otherwise in general, it'd probably sound better to use the second form. Not to digress, but it's also common in reported speech. For example "he said he would go" can mean he said "I will go", but I suppose it can be ambiguous whether he said a more hypothetical "I would go". Sometimes in speech, the "would" is stressed to emphasize that if necessary. There's probably a better way to express it though.

The second one is more accurate. Just change the word become to became though.

Here is the way that I would write this. Two elderly musicians suffered a stroke back in 1982. They both became disabled on one side of their bodies as a result.