1. By watermark do you mean the domain name in the bottom left corner? I haven't seen such a plugin but it can't be difficult to impliment.
2. What links are you refering to? If you mean off site URI/URLs then they should be fine.

For existing watermark plugins see:

Links and Downloads

1. https://plugins.b2evolution.net/?s=watermarks&submit=Search&disp=search
2. https://plugins.b2evolution.net/watermark-plugin

Direct link to most recent download 2020

3. https://plugins.b2evolution.net/media/blogs/plugins/quick-uploads/watermark-plugin/watermark_plugin_7.1.2.zip

Forum query

4. https://forums.b2evolution.net/can-i-watermark-images

But before converting, will my site links remain the same?

Yes. You can configure to use the same URL scheme. And even if you don't, all your old URLs will be redirected automatically.

See: ForeverLinks