Nope, all groups do not have to play the same machines, nor do groups have to play all machines at the location. Having more machines is a wonderful thing: less waiting between games, the game #1 machine choice is more interesting, and players don’t get quite so tired of playing the exact same games week after week. The more machines the merrier!
As for how to do it: the FSPA rules describe that the player who won their group in the previous meet and has now moved up to a higher group gets to pick game #1 for their group, and the remaining games are chosen “according to a well-defined scheme”. It’s left to the league official to select that scheme, but since you’re using the League Manager software, it offers a solution: after each group has selected their game #1, open up the scoresheet in League Manager, fill out the game #1 choices on the site, then click the “Shuffle Machines” button. It will ask you to confirm that all machines are available (e.g. at this point you could exclude a machine that is broken on this particular evening), then click Shuffle to proceed. Voila, all machines for all groups will be filled out, avoiding (as much as possible) having any group play the same machine twice or having the same machine used by different groups in the same round. In your case, with 6 groups and 5 machines, two groups will wind up with the same schedule.
Copy the machine assignments onto the paper scoresheets and send players on their way. This also has the advantage that the machine choices are already selected in the software, so you don’t have to do that when you’re entering scores later. Also, some leagues are super-ambitious and bring a portable printer to the league location, and print the scoresheets with machine selections on the spot, which saves the tedium of hand-copying the machine assignments onto the paper sheets.
Hope this helps!