I understand the intimidation of a circular saw. I would say, though, given your industriousness with other projects, Cookie, you might invest in a power miter saw. That's a 'stationary' circular saw which gives you more control and will not kick back. Angles, clean cuts will be a breeze with a saw like that.
You might also consider getting a Dremel or Rotozip. For cutting 5/8" laminate pieces, these might work ok for you. Those tools are also very versatile for grinding/sanding/polishing/trimming just about anything in a pinch.
You could always do the job by hand with a good manual miter saw. I'm talking about the ones where the saw is attached to the guide. A handsaw and the plastic BuckBros POC won't cut it (no pun intended). You gotta get yrself a nice fine-tooth blade. IMHO, though, a power miter is the way to go. Delta MS250 is a good home saw for a cool $100.
The miter saw will make short work of all the pvc yr gonna have to eventually cut to replace your pesky floor drain ...