Since you've described your feelings as anxiety as opposed to fear, I assume you recognise your feelings for what they are - something of an over reaction to the things you're getting nervous about.
It might help to both reflect on the unlikelihood of anything really bad happening to you out there, but also to prepare yourself for some possibilities such that you're appropriately equipped, and also know-what-to-do in the situation. This should help your confidence.
There are for example a number of things you could do to prepare for the possibility of running into some overly friendly doggies. Here's an article about what to do in this situation:
crazyguyonabike.com: Bicycle Touring: Dealing with Dogs, by Neil Gunton and a page where you can see the pepper-spray mounted on his handlebars :
The Art of Bicycle Touring: Handlebars: Drops vs trekking, stems, mirrors and handlebar bags. Carrying pepper-spray isn't something I'd personally consider necessary, but then I don't worry about running into dogs and you do. Perhaps if you were confident you were prepared for such a situation, your anxiety would ease.
I'm also assuming that your anxiety stems more from the prospect of going out on these country lanes rather than the reality of actually being there. If so, as you probably know there's no greater remedy than just biting the bullet, saying "to hell with it" and going out there anyway. It's not easy, but that's how we build confidence that we didn't previously have. With that in mind I think it would be great if you manage to overcome this rather than
relying on having a regular cycling partner, even if you find one. As nice as it is to cycle with other(s) it's also good to feel empowered to be able to do it yourself.