With the Rams being up against the cap for this year, how can they extend AD to pay him more for this season? I've been assuming that an extension would only pay AD slightly, if any, more than the 5th year option was already going to pay him this year, but with big increases for the next few seasons and lots of guaranteed $$s. In that case AD would be missing out on the guarantees if he doesn't sign, but will not see some lifetime negative effect on total earnings if he does not sign this year (except for a career threatening injury).
In that case, if for 2019 the Rams can't sign him and threaten to use the tag, AD could simply refuse to play and force a trade. Because in that situation his lifetime earnings could be significantly less for playing on the tag then in trying to force a trade to a team that will pay him his market value. Even though AD is the caliber of player where a team may be willing to pay 2 1st rounders for him and his new contract, don't count on it. The Rams would likely have to trade him for less.
AD has already lost ~ 50-60 mil playing on a rookie contract vs what his play warrants. I doubt he will be willing to play for less than he's worth after the 2018 season.
I think (hope) they will get something done. My guess is the Rams are working on 2 options (assuming what we've heard about a 3-year opt out clause is true). A 3 year contract that pays ~ 21-22 mil starting 2019. A 6-7 year contract that pays 23.5 mil for 2019 and goes up 1/2 mil every year thereafter and that can be voided by either the Rams or AD every year after the 3rd year (all guaranteed thru the 3rd year) if performance != pay.
If the Rams can't sign AD this offseason, then they likely will lose him, though there would still be a chance they could get him signed before next year's tag deadline. IMO, AD will not play for the tag amount. They will likely need the tag for other players, unless they all of a sudden sign all the pending FAs they want to lock up for their core. So, they will either get ADs contract done this offseason, or they better trade him this offseason if they want decent value in a trade.