I am guessing based on your 6-month admission that you are probably on B2 visitor status. If you are on B2 status, you should apply for a B2 Extension of Stay (EOS) before your current B2 status expires. If you are on some other kind of nonimmigrant status, you should apply for Change of Status (COS) to B2 before your current status expires. Either way, the process is the same -- you file form I-539 to apply for Extension of Stay or Change of Status, with a $455 fee. The application must be officially received by USCIS before your status expires, so file a bit early. You can stay in the US while your application is pending.
The I-539 application will probably take several months to process, and if you only need to stay for a few more days, you will probably leave while it is still pending, which is fine.
If you do not file an I-539 application before your status expires, you will become out of status and deportable after your status expires, though it is very unlikely you will be deported in a few days after your status expires. You also start accruing "unlawful presence" when you stay past the date on your I-94 (and don't have a pending EOS/COS application). Accruing 180 days of unlawful presence and leaving the US will trigger a 3-year ban; accruing 1 year of unlawful presence and leaving the US will trigger a 10-year ban. If you leave after accruing less than 180 days of unlawful presence, you will not have a ban, but accruing even 1 day of unlawful presence causes the visa that you used to enter the US to be automatically voided under INA 222(g), and forces you to apply for US visas only in your country of nationality for the rest of your life.
The US has no exit checks, no matter if you leave by land, sea, or air. You will not encounter anyone who deals with immigration on your way out, and there is no one to show a "letter" to. Rather, it is when you apply for a visa again, or seek entry to the US, where your past history will come up. If you stay past the date on your I-94 without applying for EOS/COS, and leave a few days after your I-94 expires, that negative history can increase the chance that you will be denied a visa (which you will have to apply for since your existing visa would have been automatically voided). You could show your "letter" to the visa officer as an explanation, but they would ask you why you didn't apply for EOS/COS, so I am not sure it would help.