Replace default widgets from functions.php file in WordPress theme

I’m creating a WordPress theme that relies heavily on the default WordPress Widgets. The default Widget markup won’t work in my theme, so I have essentially re-written the widget markup for the Archives, Categories, Search, tag wall, etc to display how I’d like it to display in the theme.

The only problem is that this only works when I’m using a “static” sidebar. When I switch to the dynamic sidebar (by enabling the specific widget(s)), it goes back to the generic markup for each enabled widget, which is not what I want. Is there any way to override the markup for widgets on a per-widget basis (Basically, is there a way to change what is output if a user enables that widget)? I’ve read on the WordPress Widget API (https://codex.www.ads-software.com/Widgets_API) that something needs to be done with registering sidebars and such, but I can’t figure it out. Does anyone have any tips?