A giyur lechumra is usually intended for someone who's already fully observing Judaism. The execution is the same -- circumcision or drawing blood for men; declaration of mitzva acceptance and immersion in a mikva witnessed by three rabbis. If someone's already keeping everything but for instance oops, found out they were adopted, then the rabbis assume the person is knowledgeable and committed, so there's no delay period for education and proof of commitment. And in some circumstances (such as populations with questionable status, such as the Ethiopian Beta Israel), some rabbis may allow a woman who had a giyur lechumra to marry a Cohen, like a born Jew, as she probably was born Jewish.
If someone was not raised observing Judaism, my impression is that most rabbis will expect a similar process of study and commitment to that undergone by someone applying who has no prior Jewish blood. Such a woman would probably be advised against marrying a Cohen. But regardless, you should contact a local rabbi -- or a national-scale body such as the Beth Din of America -- and ask.
I forgot to mention -- welcome! I hope your spiritual journey goes well. Like any major achievement, sometimes it can take some work, but it's worth it.