I was in a similar situation recently and started using W3 Total Cache too.
I’m not sure what settings you have at the moment, but it’s worth experimenting a bit if you can to see that you’re really using W3 Total Cache to the max
For example, I originally only had Page Cache and Browser Cache checked on the main screen, but I hadn’t gone into the dedicated sections and optimised them there too.
My advice would be to try enabling different options (checking different boxes) one at a time and previewing the changes, then once you’ve made a few re-run your speed checks
You can also try turning on the Minify setting in W3 and experimenting with that. You can either do Auto or if you really want to go to town chose the manual setting and then on the dedicated Minify page, use the Help option towards the top to automatically detect Javascript and other CSS files that you can potentially minify (don’t forgot to add them via the appropriate checkbox too), you can also chose to delay the loading of these files in some cases which may help with your Javascript parsing issues.
Be extra careful with Minify and test the changes (on more than one browser after clearing all caches) as you may find that some scripts don’t ‘like’ to be minified or won’t accept being loaded later in the page (for example you might botch the loading of the CSS on your site)
Finally I would also recommend trying Cloudflare (though I’ve seen people have mixed results with this in terms of speed) and also a CDN
I use MAXCDN which seems good and cheap to me and is nicely integrated with W3, here’s some more info on that:
http://www.wpbeginner.com/plugins/how-to-install-and-setup-w3-total-cache-for-beginners/
I am not an expert at this yet but I’ve used W3 Total Cache to significantly speed up a number of sites in the last few days, the key is experimenting bit by bit and testing, it can be a bit laborious but if you’re looking to get the absolute most out of your WP setup it’s worth a try
PS Also try and install the WP Smush It plugin and use it to compress all your images
PPS Always gets said but just to repeat, disable any plugins you’re not using too
Hope this helps!
Big love
Nick