Hi there,
There's no way around this - recommendations take time and your table pounders will need to dedicate a significant chunk of time to your applications. However, this is what they're signing up for and it's not that uncommon. If they've done this before, they'll understand how this works. If they haven't, they'll require significant coaching and prepping. However, either way, don't feel bashful asking those who support you to do just that!
However, let me ask you this - how many schools are you applying to? Typically, you don't want to have a single recommender write recommendations to more than 4/5 schools. 5 is pushing it. 3 - 4 is the sweet spot.
Look forward to hearing from you!
Bhavik