I’ve never heard of "ActionScript " or “Dynamic Text”. But …
A true html <textarea> object cannot support any fancy formatting. Everything in there is interpretted as (& displayed as) raw characters. Specifically, the content is NOT interpretted as HTML, so things like <i>, etc. don’t work in them.
There are alternate script-based or ActiveX-based input areas which support at least some embedded formatting. Switching to one of those may solve your problem.
But trying to stuff HTML into a <textarea> is a 100% dead end.
p.s.
When I first read your post I missed the point that you were trying to format inside a <textarea>. I thought you were trying to apply formatting to dynamically generated results in general. On that basis I wrote what follows. It may be irrelevant, but since I spent the time, maybe it’ll be helpful to you or to somebody else. …
The fact you’re seeing the markup like <i>text</i> pretty well proves the output is being html-encoded after you’re done with it.
if you’re not familiar with the term, “html-encoding” is a protective technique used by websites to prevent people from inserting malicious html or script into the results. Unfortunately, it can’t tell the difference between safe things like italics and evil things, so it simply prevents all html in the output.
It works like this: instead of sending “less-than i greater-than” to the browser which would interpret it as a signal to begin italics, the website sends “special-symbol-that-means-display-a-less-than-character i special-symbol-that-means-display-a-greater-than-character”, and the browser displays “<i>”.
And if the website is configured to apply that correction after you provide the response to it, there’s no way for you to control or counteract that behavior through the results you provide.
These things can be hard to write about on a messageboard because it too is protecting itself by html-encoding what you & I enter.
I’m assuming these tools are something you’re using on some public site like facebook, etc. If so, you’re stuck; the webmaster wants to ensure you cannot put html in your results and that’s that.
If these are tools you’re using on a website you completely control, then there might be an 'advanced" option to turn off html-encoding. Given the habit of modern consumer software to make itself “easy” to use by never using any scary (read: “technically accurate”) terms for the options, you may have a hard time finding the option you need. “Enable html”, “enable script”, or “enable unsafe output” might be examples of the terminology they’d use.