Would the IRS automatically compute a loss carryover amount if my capital loss was more than my income

If you have a net capital loss, up to $3,000 can be used to offset other income, including interest.  You don't have to file a form, the capital loss will flow from Schedule D to Form 1040, page 1 line 7.  Any loss above this amount will carry over to the subsequent year until used up or offset by capital gains.  The calculation of your capital loss carry over will be verified by the IRS when you file your tax return.  

No - you need tt premium for selling stock and fill out form 8949. If capital loss is greater than capital gain you can offset a maximum of 3000 against other income for this year {2024} and if you still have loss you can carry it forward to next years (2025) taxes until the loss is used up.