Somebody bought my old domain and I can't access my old mail

"I don't have access anymore to the email account that was used to sign up" is not a rare occurrence.

People commonly use their work email and will lose access to that once their employment ends is a very common scenario (and possibly easier than explaining an expired domain name) .

Depending on the nature of those two critical websites you can of course try to contact them. They may be willing and able to help you using an alternative ID check, especially if the email address(es) used are currently not valid anymore.

What @tater already commented:

Talk to the new owner.

You have two favourable outcomes:

• the new domain owner has a business model of buying expired domains and is willing to sell you back your old domain (at an arguably inflated price)
• the new owner is willing to lend you a hand and will accept and forward the account reset messages on the email address(es) you had in use. (And you send an Amazon or similar gift card as a polite thank you if they don't bill you for their effort.)

Be aware at what you disclose as there is a potential unfavourable outcome as well:

• the new domain owner is malicious and tries to use their current control of your old domain (to use the e-mail password reset process) to gain control of your accounts at those two critical websites and possibly others.