Seriously...

Offer what they want... no coach.
Offer longer deal... no coach.
Offer shorter deal... no coach.
Offer less money... no coach.
Offer more money... no coach.

Gotta outbid the other guys

They don't evaluate offers during the draft, if you're in a league that's in the middle of the draft. If they reject your offer then they did it to sign with someone else for a more money per year. Unlike players, a coach will take the best offer they have (per year) during any free agency, midweek, or game stage.

If there are a lot of vacant coaches then it can be tough to get one. If you enter a game stage without a coach the AI will choose the best available coach and offer him a contract, so you won't go into a game coachless.

Hmm... kk, makes sense.

If your coaches are under contract, can you promote and demote your own coaches without firing them?

Yes, you can. However, it will fire whoever is currently at that position.

EDIT: I actually don't know if you can demote a coach since I haven't tried. But lateral moves (ie QB Coach to OL Coach) and promotions are both possible.

You can promote, but you cannot demote without first firing. Ares is correct that promoting _will_ fire the coach in the position that you are promoting him to.

So how do you do it because I can't find a tab that would allow me to do this? These coaches are under contract for a few more years.

If they signed a contract that same year, they are unavailable for re-negotiation. Otherwise, on your team page, just select the 'Coaching Staff' tab on the right panel, select the coach you wish to submit a contract offer to, and on their pop-up card there will be the option for 'Contract'.