I ran your homepage through the Shopify Analyzer (free analysis tool our team built for the Shopify community) and it's actually scoring pretty well (91/100). You can check out your full report here if you'd like: https://analyze.speedboostr.com/result/g0nunn73fo. On desktop, it does look like your banner/slideshow images are a little oversized. I'd recommend shrinking the overall size of the slider and reducing the size of your banner images to 1400px or less.
I also navigated through your site and everything else loads pretty quick on my end.
Regarding all your products having NEW or SOLD OUT badges on them. Is it possible this is functionality coming from an app, or have you had any developers work on your theme? It looks like all of your Collection images have a class on them that either specifies it should have a NEW badge or a SOLD OUT badge.
I can't be sure without directly going into your code, but it looks like there's either a piece of code in your theme that determines what badge an image should have or it's an app performing that process. If you haven't had any developers work on your theme then this is more than likely coming from one of your apps.
Sorry I couldn't be of more assistance.