You can't convert without him finding out. A Rabbi won't let you go through the process without it fitting into your life.
It's time to do some soul searching. Maybe start with a good therapist. (There are plenty of so-so ones, so look until you find one that feels right & HELPS you sort things out without judgement of you.)
If you are better off without him, you've answered your own question. You have a new path in front of you that feels right, & an old path that doesn't. (both in relationships, & religion) However, you have to resolve the relationship one, and see who you are, before you start another big change.
You can talk to a Rabbi about this. Remember the Rabbi's job is to discourage you from converting. He'll want to make sure you are committed, ready, & it's right for you. It's okay to take time on this whole pathway. Life is a journey. Judaism is a journey, so you'd be beginning your Jewish life even at that level.
Consider telling your husband. If might go better than you think. If not, then consider that a marriage should support & enhance your life -- not restrict your valves by someone else's.
All the best with this journey. Maybe say a traveler's prayer, take a deep breath, & get started?
To clarify on vassesm answer:
Other than Orthodox, all the branches would consider you married. Orthodox may vary from group to group, but i'm much less familar.