A correction to previous information in this space: Mark Fishman recently told me that "at the end Roy did update and print brand-new complete manuals for the operating system(s) and BASIC(s) -- one manual covering both LDOS and LSDOS, and one for both MI/III and M4 BASIC." I don't have copies of those manuals, but I hope to get them eventually.
LS-DOS 6.3.x was an upgrade to TRSDOS 6.2, whose manual was published by Tandy. You may still be able to get these by mail from Tandy Software Replacement (Radio Shack Unlimited), but I wouldn't count on it.
LSI supplied update pages with LS-DOS 6.3.0 and 6.3.1, detailing the changes since the TRSDOS 6.2 manual was published. These are now online. There was also a Tandy-edited version of these pages, but I currently don't have a copy.
Most questions about the operation of LS-DOS 6.3.x can be answered using the extensive online help system included with version 6.3.1. Quite a bit of the information in the Model I/III LDOS 5.1.x manuals and 5.3.1 update booklet is applicable too. Complete technical information for programmers is available online in the Programmer's Guide, and the complete source code is online too.