Lets create special offer type "sold" that will hide properties from the search form, but make old links still works (better for seo / no 404 errors).
1. Login to your admin panel
2. Hover Properties and click "Offer Type"
3. Add new offer type Sold with e.g. black background label and white font color
4. Check the box ""
5. Click Add New Offer Type
Now when you assign this offer type to your property it will be not displayed in the search form.