Your question in no way silly.
First, you must deicde whether you really want to compete. You can work at your dancing and enjoy it for its own sake. I think all competitive dancers enjoy dancing - whether it's working hard to perfect something, or just dancing socially, or competing in a major competition. Personally, i love to dance, will often dance around on my own in private to just let my feelings come out with the music (and the better my dancing becomes, the better this feels). But i also love to compete in major competitions because, like you, i have always had an element of competition within me.
If you really want to compete and you feel this is an element missing from your dancing, then you should look for a new partner. But first tell your present partner how you feel about things and that you have this 'need' to compete and will find a new partner, Try to say this in as friendly, and non- confrontational way as you can. Also, and importantly, ask your partner why she won't compete (you didn't mention any reason). Is she just shy (that's very common) or is there another reason. Perhaps when she realises you may look for someone else, she will make an effort at competitions.
Don't secretly look for another partner 'behind your existing partner's back'. Quite apart from giving you a bad reputation, it is completely disrespectful to her - she has spent a lot of time working with you on the dancing and presumably is at least a close friend. Always remember, your partner is another person with his/her own set of feelings.
Good luck with your dancing