As I say in the Comment above, it is possible that just sanding deeper would resolve the problem with splinters1. It depends on the level of weathering whether this is viable; you don't want to sand too deeply and change the profile of the handles too much and/or weaken them, plus there's no non-destructive way of knowing in advance if it will work.
So the other option that first occurred to me was to consolidate the wood that's there. This can be done in a number of ways but essentially, or literally, you will be glueing the splinters in place.
Coat them in something
- Exterior varnish. Spar varnish1 is a type of exterior varnish but intended for more fully exposed contexts, it's softer and may get slightly tacky with use in hot weather.
- Paint. A suitable exterior paint could be enough to do what's needed, although I'd expect varnish to work better (since varnish is "all binder" so to speak).
- Epoxy. Some might suggest a penetrating epoxy (CPES) here but again cost may be prohibitive, and it may not work as effectively as just straight epoxy adhesive anyway.
Cover the ends with rubbery grips
I'm familiar with these from metal wheelbarrows so they immediately came to mind as a possible solution. I didn't know if they were available to buy separately but I tried a search and they are, not really surprisingly. No idea if any are made that will fit your handles but worth investigating I thought.
Obviously you could just use duct tape or something similar here, although the grip wouldn't be great many a tool handle has been fixed or bulked out with duct tape! There are grip tapes though which would probably do everything needed here, prevent splinters forming and improve on the current grip so win-win.
Wrap in string, cord
If you don't use a synthetic you can protect the natural fibre if necessary by overcoating in varnish or epoxy.
And last but not least....
Do nothing, just wear thick gloves
1 Underneath the surface of weathered wood there is very frequently sound wood, and sometimes surprisingly close to the surface (although I expect that's not the case here).
2 A true marine spar varnish is probably overkill, and also an expensive purchase for just the one utilitarian item. Plus you probably won't find any for sale at the usual places you might be looking to purchase anyway.