Well your best bet is to try both and profile them.
My guess is if you are using a single 4096 length image (max size for a texture on most cards) then there is no point cutting it up. The gpu will clip the polygon to the screen anyway so there should be no performance penalty at all.
If however you are already using multiple textures to get a length greater than 4096 then it may be beneficial, though more in terms of memory management than performance.
As for your lines where there are joins you should check that texture repeat is set to clamp and maybe use point filtering.