You can keep Namecheap as a registrar but use a different DNS provider.

I have Namecheap for registrar and Cloudflare for DNS, another provider with solid free service is NS1.

At that point the only interaction in terms of DNSSEC you have with your registrar are managing the DS records with your parent zone. Namecheap fortunately does this with a web form, whereas I've seen others require it through email/support ticket.