Computershare is fairly easy to deal with. They are the administrator for numerous dividend reinvestment and direct stock purchase plans. Years ago, these low fee plans were the least expensive way for an individual to buy stock. The company, not Computershare, sets the fees. You can look up the fees for XOM on the Computershare website.
You can keep the stock in Computershare, have them sell the shares for their fee, or you can transfer the shares to Fidelity. Transferring the shares is not a sale. If you transfer the shares, there will probably be a transfer fee. Any partial share will be cashed out. Fidelity will also reinvest the dividends and they will do it at no charge. You do need to track the basis, which is going to be a big PITA because you probably don't have the records.
I assume the account is in your name now. If not, that will have to be cleaned up first.
I still own some XOM shares. They pay a fairly low dividend for the industry, but I'm not inclined to sell and pay capital gains tax on the appreciation. Like you, I have been ignoring it for years and I will probably let the heirs figure out what to do with it.