Jag är trött, men jag måste jobba ändå.

Because jag (I) refers to one person, so the adjective takes the common singular form: trött.
Trötta is used for plural subjects (vi är trötta = we are tired) and also as the definite form in some contexts (den trötta mannen = the tired man), but not with jag.

Men means but and signals a contrast: I’m tired, but…
Och means and and would merely add information, without the contrast. The Swedish sentence is emphasizing the conflict between being tired and still having to work.

Swedish often repeats the subject after men when it starts a new clause: …, men jag måste …. It’s the most natural and common phrasing.
You can omit it in some informal speech/writing (Jag är trött, men måste jobba ändå), but it sounds more clipped and is less typical for learners to model.

Måste means must / have to—a strong necessity. It can be:

• obligation from outside circumstances (Jag måste jobba = I have to work)
• strong personal necessity (Jag måste ringa = I must call)

It’s generally stronger than English need to and close to have to/must.

After modal verbs like måste, Swedish uses the infinitive without att:

• Jag måste jobba (I must work)
  Not Jag måste att jobba.

Att is used in other infinitive constructions, e.g. Jag vill jobba (no att after modals like vill either), but Jag försöker att jobba / Jag försöker jobba can vary depending on verb and style.

Jobba is the verb to work.
Jobb is the noun a job / work (countable in Swedish):

• Jag har ett jobb = I have a job
• Jag jobbar = I work

So the sentence uses the verb form.

Ändå means something like anyway / still / even so, adding the idea “despite what was just said.”
Placement is flexible, but common positions are:

• end position (very common): … jobba ändå.
• earlier for emphasis: … ändå måste jag jobba. (more marked/stylistic)

End position here sounds natural and conversational.

In Swedish, it’s standard to use a comma before men when it connects two main clauses (each with its own subject and verb):
Jag är trött, men jag måste jobba ändå.
You’ll sometimes see the comma omitted in informal writing, but the comma is a safe default.

After men, the clause is a normal main clause that is not starting with a fronted element, so Swedish keeps subject before verb: jag måste.
You get verb–subject order (V2) when something other than the subject comes first, e.g.:

• I dag måste jag jobba. (Today, I have to work.)

Here, the clause begins with the subject jag, so it stays jag måste.

Yes. Arbeta is a more formal/neutral verb meaning to work, while jobba is more everyday and common in speech.
Both are correct; jobba fits casual conversation best.

Trött is broad and covers both general tiredness and often sleepiness depending on context.
If you specifically mean sleepy/drowsy, Swedish can also use sömnig (sleepy), but trött is the most common general choice.

A common spoken emphasis pattern is:

• Jag är TRÖTT, men jag MÅSTE jobba ändå.

The contrast is often carried by stress on trött and måste, and ändå can get stress if you want to highlight the “despite that” feeling.