Protecting your investment is your primary concern. You'll likely go through all the same channels and jump through all the same hoops as buying a home. Both homes will need appraisals, inspections, title work, etc... Why? You'll want to know that you're getting a legitimate home in exchange for yours. In a trade it's not about the "price", there's concerns of judgements, liens, encumbrances, maintenance, etc.
However, if you have a home with more square footage, and is of greater value, wouldn't it be better to sell it and have extra cash in the bank when you purchase? Also, marketing your home properly will ensure its exposure to the world. Is your agent doing everything they need to procure a buyer?
I know it can be done, but is it the best thing.
Bear with me as I use my dead fish analogy. You have a dead fish (home that will not sell on the open market) and you want to trade it for another dead fish (home that will not sell on the open market), if you find someone for your dead fish. Likely you will not like theirs.
I know there are on-line sites for trades and that is where I would start. Also it is a good idea to have 2 appraisals on both properties you can agree to split the value in the middle of both appraisals. This will help you from trading to low.
Let me know if you would like to chat.
best of luck