What you might do is tin just the very tip of the wire, not the whole thing down to where the insulation is stripped back. This will eliminate the “strays” and still leave the flexibility, if you need it.
Use solder intended for electrical/electronic work, typically 60 tin 40 lead rosin core.
Do not use acid core solder or soldering paste. Solid solder may work also but it is more difficult to work with because without flux (rosin) it won't tin the wire as readily, and lower tin mixtures and non-lead mixtures require a higher temperature soldering iron.