However, its no longer an adult site, and now suitable for all ages. Google/Bing keep looking for URLs from the old site. I made sure they all 404, but is there a better procedure? Should I 301 those links to a page explaining that there was a change of ownership?
Absolutely do not 301 redirect any old link to the site. The assumption would be that the link and page are valid and they will remain in the search engines indexes as a result. 404 all of these pages and let them drop from the index. It will take quite a while. As well, search engines will continue to follow these old links as long as they exist. There is nothing you can really do about that except to see if you can get the links removed. I would not even advise disavowing these links. Just let the chips fall where they may and do your best to create a high quality site and time will reward you. Do your job right and it all will work out okay- it just will take time.
On a side note, I always advise against picking up domain names that have been previously used for sites that are off-topic for your wishes especially sites with bad histories. It is always an up-hill battle that works against your sites performance. The degree to which can only be determined on a case by case basis. I always recommend staying away from previous adult sites unless you want to run an adult site. Yes you can get traffic from an established domain name, but is it the traffic you want and can convert? As well, any domain has established trust issues that have to be overcome and that is the foundation upon which SEO is based. All other efforts are moot if the domain has low trust, safety, child safety, or other performance scores that will take a year or more to overcome.