No, you're not required to sign any contracts. But since you're offering a service, you do have to manage some compliance tasks.
You are a data controller under the UK GDPR, regardless of whether you have a company. This brings with it various compliance requirements. For example, you MUST post a privacy notice in accordance with Art 13 GDPR that explains how you process user's information. You must sign data processing agreements with your data processors, such as your server providers. And you should reconsider transfers of data to the US, since such transfers are illegal or at least quite questionable in the wake of the 2020 Schrems II ruling.
You now have prospective users that are asking you for an Art 28 data processing agreement (DPA). These are likely other organizations that are data controllers of their own. For them to use your browser extension, they either need to find a legal basis that allows them to share their user's/employee's data with you (controller to controller transfer), or they need you to act as their data processor (controller to processor transfer). The third alternative is not to use your plugin at all. Of these, a C2P arrangement is most convenient for these other orgs, but involves a bit of paperwork to set up first.
If you want to act as their processor, this doesn't mean you'd have to create a company (though a corporation might be very desirable as a liability shield). Being a processor means that you're contractually bound to only use the personal data as instructed by the controller, and not for your own purposes. This restricts what you can do, and has some special compliance requirements. For example, you cannot engage new subprocessors without your controller's approval. However, processor status can also be convenient for you because you're not responsible for interacting with the controller's data subjects, e.g. you're not responsible for handling their data subject access requests.
Note that it's possible to simultaneously be a controller for some users, and a processor for others. E.g. Google Docs is offered directly to users as a B2C product so that Google is a controller, but also as a B2B product as Google Workspace, where Google acts as a data processor.
Personally, I'd rather not sign any contracts unless I'm doing it as a business, and adequately compensated for providing this service. GDPR is only one compliance aspect out of many, making it unwise to offer a SaaS product as a hobby. For example, copying other people's web content has copyright implications…
A potential alternative for some of the organizations asking you might be to make it possible for them to self-host your backend, so that they are not required to rely on you as a data processor. If you have no plans to commercialize your software, making it Open Source could be a solution. But you're in no way required to do that if you don't want to.