I don't think there is a quick fix for the D - short of Quinn Hughes being the second coming of Scott Niedermayer as a rookie.
There are incremental steps. Firstly, addition by subtraction. The weakest points on the D are MDZ and Pouliot; both are free agents (UFA/RFA) this spring. It would make some sense to move one or both at the deadline for a mid-round pick. It's possible Gudbransson could be moved, but with 3 1/2 years remaining at $4M per that is a difficult task.
Addition by subtraction only works when you have someone else ready to take those spots / minutes. Olli Juolevi was progressing in his first season in the AHL, but the ankle injury was a set back. I'm not sure if he will be ready to take regular NHL minutes down the stretch. He was playing high minutes in Utica and generating offense, but his defensive coverage was not great. If he progresses to the point where he can play 3rd pairing minutes as an NHL rookie without hurting the team, it will allow them to move MDZ and Pouliot.
Quinn Hughes is going to get some games towards the end of the season, but who knows when his Michigan season will end?
Next up on the depth chart in Utica are Jalen Chatfield, Guillaume Brisebois and Ashton Sautner. Are any of these an improvement over MDZ or Pouliot at this point?
So where does that leave us? College free agents, European free agents, a trade or the summer UFA window.
I don't think an Edler or Tanev trade is in the cards, and honestly, we would be lucky to find a UFA defender as cheap as Edler or Tanev who are anywhere near as good. IMO the most sensible approach is to re-sign Edler if he will sign a shorter term (2 - 3 year) deal.
I expect there is a Sam Gagner trade in the future. Granlund and Baertschi are also potential trade bait. Granlund is a decent utility player who can any forward position on any line well enough but on a good team should not be a regular in the top 6. Adam Gaudette will fill this utility role going forward, and he would provide cheap utility without a long term commitment to a playoff team with injuries. But none of these moves will garner a top 4 defender under 30.
Tryamkin? I think he still has 1 year remaining on his KHL deal and even if he voided that deal and signed with the Canucks in the summer. After 2 years in the KHL he may not be the same player we saw in 2016-17. He has also shown signs of being a bit headstrong and a challenge to coach. IF he comes back to the NHL and IF he is an effective player I would count it as an unexpected bonus.
Edit: Improved goal tending will also help the D. It really hurt our D when Nilsson went down and Green played Markstrom for a ridiculous string of consecutive starts. Both look better with some rest.