How can I improve my LL without learning full OLL?
If I can improve my LL to about 4 seconds, I can be sub 16...
If you can already turn to a good tps (say, 5-7), then the only real ways I see that you can improve on is:
1) learn da OLL's narb!!!! But you obviously don't want to.

2) Improve your algs. Go through each alg and look through the wiki if there's better one (if you haven't already)
3) Just let time allow you to improve it. This is what I did. Simply practicing allowed me to recognize cases faster and I naturally developed better fingertricks without me even knowing it.
PLL time attacks work for some people... but I've never seriously done them.