What I would suggest is that you go to the plugin settings page, and disable the first option under Plugin Styles (Load the main stylesheet? (colors and layout)). At that point, you’ll have just buttons with icons, and all styling for the buttons (other than the icons) will come from your theme stylesheet. At this point, you can adjust the CSS however you like.
If you need a starting point, you can copy the CSS from the plugin’s scriptlesssocialsharing-style.css file (in the includes/css directory) to your theme’s stylesheet, and then change as you like.
Hope that helps you get started–
Edited to add: the plugin default styles do actually behave as you are describing in your question, now that I read it again, so it’s possible that your theme is already overriding the plugin styles (some themes have !important in their .button CSS). You may want to leave the plugin styles as is and investigate what’s going on with the theme. You may just need to override styles for these with more specific (and possibly !important) rules.
Thank you. Your addition is probably what the issue is: My theme has all links to be displayed as blue. This is probably what is overriding the plugin’s style sheet. I will investigate overriding this.