The accrued interest you pay at the purchase date is not an adjustment of the bond premium and is not included in the amortization of premium or discount. It is returned to you in full at the next coupon date that occurs after your purchase date. You claim the accrued interest as a reduction of your taxable interest income at the time of the next coupon payment by entering it as a negative amount (identified as "accrued interest") on your Schedule B.
Accrued interest works the same way for tax-exempt bonds (because some are taxable at the state level and tax-exempt interest income also affects the taxability of social security benefits.) You deduct the accrued interest paid at purchase from the tax-exempt income total on your Form 1099 to arrive at the amount that should be entered on line 8b of Form 1040. You can reconcile this for the IRS by attaching a schedule or statement to your return which shows the total tax-exempt income reported on your Form 1099 and then shows the deduction for accrued interest to arrive at the line 8b amount. Check to make sure that the total tax-exempt interest income on the Form 1099 has not already deducted the accrued interest before reporting the amount to you--some brokerage firms do it automatically.
Another likely area of confusion is that the accrued interest paid at purchase is deductible based on when the bond next pays interest, which is not necessarily in the same tax year as when you bought the bond or note.
At the time of sale, the accrued interest that you are owed for the period from the last coupon date to the date of sale is paid to you by the buyer. It is usually automatically added to your interest income total by the brokerage firm, and you do not have to make any further adjustments on your tax return.