"I drank too much at the party."
'Sugi' is the noun and 'Sugiru' is the verb. Both are used in this lesson ("Sugi desu" & "Sugimashita").
It is really more complicated than that. "Sugiru" is a verb that translates to a range of English verbs with the general sense of "going beyond." As a suffix "--sugiru" has the sense of "doing to excess" and attaches to the same form of the verb as "-masu." (nomu + sugiru = nomisugiru, "drink too much"). This is a straightforward compound verb and "--sugiru" can be attached to just about any verb with which it makes sense, e. g., tabesugiru.
Yes, "sugi" is used as a noun but you generally will not find it in a dictionary because it is actually the conjunctive stem (renyookei) of "sugiru." The use of the conjunctive form as a noun is not uncommon but it is not always idiomatic. You have to know which verbs are used this way.
The present form of the verb is すぎる. すぎ is also a noun form that means "over-" (e.g. たべすぎ = overeater).
Why is it すぎました while on other sentence 'eat too much' is translated as 食べすぎです?
食べ過ぎ means "overeating" as in the noun. So if you said "I ate too much" it would also be 食べすぎた.
Im not entirely sure, but 'wa' is implied for the subject, while the object is the party, which is why party gets the attached 'de'. This is at least my reasoning for how i got it right on the first try.
は does not imply that the preceding phrase is a subject. は sometimes marks a noun as the subject, but most of the times it is used as either contrast or setting the topic of the sentence. The は in パーティーでは is used for either marking contrast or thr topic, but never the subject, as it is a phrase rather than a noun.
Even in English "at the party" is not an object. It is a clause complementing the verb. Same for Japanese that パーティーで is a complement rather than an object. で marks the location where the action takes place.
Now パーティーでは飲み過ぎた translates to "At the party, I drank too much." "At the party" should go in front of the sentence if は is used to make it the topic.