Learn Romanian: Am așteptat o săptămână întreagă.

You can, but the grammar changes. Întreg (and its feminine form întreagă) means "whole" or "entire" and pairs with the indefinite article: o săptămână întreagă (a whole week). If you want to use toată (all), it takes the definite article and goes before the noun: toată săptămâna (all week).

Why is the past tense am așteptat used here instead of the imperfect așteptam?

Even though waiting for a week is a continuous, drawn-out action, it is presented here as a completed event with a clearly defined duration. When you state the exact duration of a finished action in the past (like o săptămână întreagă), you use the compound past (perfectul compus).

Why does the adjective întreagă have an "ea" in it when the dictionary form is întreg?

This is a very common vowel change in Romanian. When a feminine noun or adjective ends in -ă, a stressed e in the root often changes to the diphthong ea. So, the masculine întreg changes to întreagă to agree with the feminine noun săptămână.

Can I put the adjective first and say o întreagă săptămână?

Yes, you can! In Romanian, putting a descriptive adjective before the noun gives it special emotional emphasis. O săptămână întreagă is the standard, neutral way to state the fact, while o întreagă săptămână emphasizes your frustration or shock at just how long that wait felt.