Having just run into this same issue...
HTML signatures converted to plain text will have additional asterisks
This is when you use <b> tags and the like ; Thunderbird boils them down to the same markup you can use in plain email for emphasis that it will render.
What you can do is use HTML style features that Thunderbird does not do this with ; use CSS instead of tags.
e.g.
Instead of
<b>This bit is in BOLD</b> Use
<div style="font-weight:bold">This bit is in BOLD</div> Doing this I found it reasonably easy to create an HTML signature file that rendered down nicely as plain text. You can even include images - if you edit your signature by editing an empty email and saving it as HTML. Drag your image into the mail from the file manager, rather than using Insert > Image, and it will be embedded as base64 rather than linked to the file, enabling you to create a standalone sig file that can be shared and edited by other users.