Learn French: J'allais faire la lessive, mais je suis vraiment débordé ! - I was going to do the laundry, but I am really swamped!

Using the Imparfait of aller followed by an infinitive (J'allais faire) is how you say that you were going to do something. It perfectly captures an intention you had in the past that you usually didn't end up fulfilling. If you used the Passé Composé (Je suis allé faire), it would mean you actually went and did it!

Could I say laver les vêtements instead of faire la lessive?

People would understand you, but faire la lessive is the standard, natural French expression for the chore of doing the laundry. You might also hear laver le linge (to wash the linens or clothes).

What is the literal meaning of débordé?

It literally means overflowed or spilled over, coming from the verb déborder. In everyday French, saying je suis débordé is the exact equivalent of saying I am swamped or I am completely overwhelmed in English.

Does débordé need to agree with the speaker?

Yes. Because it acts as an adjective describing the subject (je), it must agree in gender. If a woman is speaking, she would add an extra -e and write débordée. This spelling change doesn't alter the pronunciation at all.