If I get it right, the next button is CTA of the page when user fill the inputs manually. so I prefer putting it in a same place for every page. It's easier to remember for users.
The right one is much better if there is no touch keyboard overlays the page.
If long text buttons is not an option for you then your can opt for other options.
Instead of button you can go with plain hyper link or an alternative to text can be an Icon image, but at times icons does not looks self explanatory, to tackle that problem you can add a tool-tip to the icon.
Option 1: Keep only Login in the4 navbar
A common pattern that is used across the board is the use of only the Login CTA in the nav bar. Once in the login screen, the user is given the option to Sign Up or Register if they didn't have an account.
Option 2: Keep both, but don't make them buttons
Keep both the Login and the Sign Up in the nav bar ...
You could use the F pattern (Arrange the information from left to right in the upper lines and then less information on the next lines)To not make it pop too much simply use the colors of your brand. You should get one contrast color from there.
Also, I felt a little bit lost in your page, right now you have two buttons that fight too much with ...
On mobile devices, I've noticed also a horizontal slide to show the side menu (same can be done for vertical ones, but you would need somehow to visualize that feature for the user e.g. with subtle glow animation or something from the top of the screen).
Also, in material design I've seen the floating bubble/circle to call the menu.
Maybe you don't want to buy something in the same product category, with that purpose I would take the user to the front page (that should be the one that let me see the general categories).
If you are really buying objects on the same category, there are other alternatives before clicking to the "continue shopping" button, for example:
"The users that buy ...