Barksdale and Blalock would fill the biggest needs on the entire roster and at this point in free agency, it is more about filling needs than talent upgrades.
With Barnes returning and with Jones and Rhaney offering the most promise from the group of the untested backups, center is not a major need (IMO). Wiz has had the recent shoulder surgery as well and I think his current asking price is too high.
The Rams currently have 9 linemen under contract, they are thinnest/weakest at the tackle spot IMO so Barksdale being a known quantity fills the team's biggest need. I think Blalock actually offers the best value as a short term upgrade plus he fills the team's second biggest need.
Primary Tackles: Robinson, Reynolds, & Baker
Others that have played tackle in regular NFL games (not preseason) - Saffold & Bond
Others that could project to tackle - Jones & Washington
Primary Guards - Saffold, Washington, & Bond
Others that have played guard in regular NFL games (not preseason) - Reynolds & Barnes
Others that could project to guard - Jones & Rhaney
Although Robinson has played Guard he is exclude because he is THE left tackle
Primary Centers - Barnes, Jones, & Rhaney
If the Rams feel Barksdale and/or Blalock will probably remain unsigned through the draft then it could be better to actually wait until after the draft to sign either one of them (if needed) depending on how things shake out.
Although Jim Thomas has indicated in his chats that the possibility of bringing back Jake Long is dead, I think he might also still be an option as well when late June rolls around (depending on need of course).