TL;DR Yes, learn to touch type, but don't learn Qwerty. Learn Colemak. (There are others, but I find Colemak to be the best option for many reasons.)
How many words per minute can you type now? Touch typists can generally ramp up to 80wpm to 130wpm, so increased productivity is one of the major reasons to touch type.
But the reason I chose to answer this is to try to convince you to change your keyboard layout before you learn to touch type. Since you are learning to touch type, switch to Colemak keyboard layout. You will save yourself a lot of wrist strain in the long run.
Colemak is designed around a few ideas.
First it keeps common key combos the same as Qwerty, so ctrl-C and ctrl-V that you have learned from muscle memory continue to work.
It takes the most commonly used letters in English and places them under your fingers on the home row.
Fingers should travel the least distance possible
Most of the typing should be done on the strongest and fastest fingers
Typing should alternate between the hands for greater speed and comfort
Should be easy to learn and switch from QWERTY
is a blog post from someone that was in the same situation as you, they didn't touch type, and started with Colemak. They typed at 75wpm in two months and 90wpm after 4 months.
One of the benefits over Qwerty are you can type 35x more words using only the home row on Colemak. To be clear, on Qwerty you can type 195 words without stretching your fingers from the home keys. On Colemak, that number is 2546 words. Here
is a listing of those words, as well as comparing the lists of Dvorak and Workman layouts, both of which offer more words than Colemak on the home row with some trade offs.
Colemak is designed to allow you to alternate hands as much as possible to reduce stress on your fingers, as well as to improve speed.