The first example has all the required information but is difficult to read. The second example doesn't parse properly at all:
Now she is going to be late to Shelly’s cousin's, welcome home party.. Landon is back from Iraq on a month leave.
The comma after "cousin's" doesn't make sense. Cousin's what? House? Also, ".." isn't proper puncuation. An ellipsis has three periods (...), not two, but doesn't belong here.
The goal is to convey information clearly. Instead of throwing all this into one sentence, try breaking it up:
Now she is going to be late to the welcome-home party. Shelly's cousin Landon is back from Iraq on a month leave, and Shelly is throwing a party for him at her house.