Ok so here is how I understand the options
Let him walk as a free agent, you can't trade him now since he isn't under contract.
Franchise tag him at either WR (11.5m) or TE (6.7m) if the Saints try and tag him as a TE, he gonna be pissed and might end up in the court of the NFL/NFLPA. Once he is tagged he either plays the season at those tags (which the Saints can't afford) or they have until training camp basically to work out a longterm (cap friendly deal).
Once tagged he can be traded only if he reaches an extension with the new team. (I think this is what changed in the new CBA), the Saints would be on the hook (cap hit) for his signing bonus. Just like when any player is traded, the signing bonus is spread out over the length of the contract comes due when traded. The Gronk contract had a 8 million dollar signing bonus, so that would be the Saints cap hit this year if tagged and traded.
Now a team trading for him, might be willing to give up more because their cap hit wouldn't be a much. But they also know he might be a FA next year and then his value might be down a little because of the big game performances this season, same reason you guys are willing to move on.
I cannot see a team paying Graham 12 million per year plus giving up 2 - 1st round picks, when there is a good chance he will be a FA next year.
There are 3 different types of tags, Exclusive (cannot sign with another team), non-exclusive (can sign with another team, original team gets 2 first round picks back) and transition (allows the original team to match the offer)
Very interesting situation
The two sides are several millions of dollars apart in contract talks, according to Fox Sports. The Saints are willing to make Graham the highest paid tight end in the NFL, but Graham is looking for roughly $12 million per season, according to the report.
Gronk is 6 years 54 million