Plans can have a way of changing, life sometimes simply unfolds...a journey of a 1000 miles begins with a single step.....and all that philosophical stuff..
$20,000 seems a little lite..but it's a place to start as you research what's out there that's potentially capable of doing what you want to do.
Sailing magazine, just did an article on the Pacific Seacraft Flickas....here's one.. 1981 Pacific Seacraft Flicka 20 sailboat for sale in
After you find a sound vessel the rest is up to you..anything is possible!
Gender is a non-issue imo.
I think the only gap might be whether or not single handed blue water sailing is a nice dream...or will it become a reality for you. If you've never sailed alone...and have never left sight of land alone on your own vessel the reality of that experience may offer a new or different perspective, or it may re-affirm your goals.
I might continue on your plan of gaining more experience in a livaboard training environment, then follow your thought of hitching a ride on an offshore trip or trips with an experienced crew....Then decide if your plan is reasonable for you before making a purchase.
Boats in that price range capable of blue water cruising can eat up savings accounts in repairs and maintenance before you leave the dock. I'm not saying your plan is unrealistic; rather, just continue your research, obtain more experience and kick a lot of tires...while you do that.