First off, getting your site mobile optimized is smart.
Next, you need to have a clear understanding of where all this traffic is coming from. If it's organic, dig into Google Webmaster Tools & even Semrush to determine how much of it is from branded queries. If a large percentage of your traffic is from non-branded organic searches, I would be very cautious of changing URLs & title tags - unless your organization is completely okay with the possible ranking & traffic loss that will most likely come with all the necessary 301 redirects. The current URLs are already far, far cleaner than a lot of sites (especially ecommerce sites that have crazy dynamic URLs).
With all that said, I have worked with a few brands in similar situations where we decided to move forward with a site overhaul (which is what you're describing) because (a) most of their traffic was either direct or branded organic and (b) they worked with an SEO agency years ago that implemented a number of spammy tactics that we needed to clean up. Ultimately, we decided that the benefits of sort of "getting straight with Google" was worth the possible ranking & traffic loss.
I hope this helps!