As someone else stated, you can be married by a notary or the State and then have a personal private ceremony. The other option for more public affairs might be to have someone from the UU officiate as they are able to bridge the gap between multi-spiritual needs of both the bride and groom, respectively.
But, in essence, I think you will find in many states that although a religious official might be able to perform the marriage it must be notarized etc. and that is the actual "official" marriage. In my case, our marriage cert was notarized and then we were married by comm unity ledger which is where all attending ppl swear to uphold, validate, and basically witness the marriage.
One other thing to keep in mind, especially with Lughnasadh being so close... In basis a handfasting by and of itself is a trial marriage and not necessarily that dissimilar from the modern experiment of living with someone for a year or more to make sure you don't kill each other. The main difference is that it was just made "official" to the community. In older occurrences, children might even be handfasted as a way to promise them to each other and secure agreements between two different groups of people. Marriage, though, is the formalized mostly permanent bond that is secured between two people - so handfasting and then later a formal declaration of permanancy. In Ireland, there used to be SEVEN different kinds of marriage and even the right to dissolve the marriage if desired. Basically, if the other marriage partner did not pull their weight and live up to their obligations it was grounds for divorce.
Of course, all of that is probably about as clear as mud, but I just wanted to say that there is alot of room for variation. Have a handfasting and refer it to the cowans as an engagement party. Then, later on, (plus this gives time for planning) hold a marriage ceremony and formalize your contract. Remember (even though this thought can be upsetting to some.. but others will understand) you can love someone and even bear their children... but when it comes to marriage it is a business contract and the formalization of the unity between two groups of people. Examining the nature and meaning of your relationship will add clarity and perspective on how you should proceed.
Plus, there is nothing wrong with getting your marriage license and then the two of you going off into the woods to recite your vows and then returning and hosting a reception/festival to commemorate your joining. Sometimes things can also be kept private and what is shared within your hearts is personal. Lots of different ways to proceed.