I hope not not worth the money plus the pick
I think if you're going to give up a pick, you might as well do it on a long-term contract. That doesn't mean give extra years just to justify a signing, but giving up the pick is more palatable when you'll have the player for several years. With a protected 1st-round pick plus a supplemental from Baltimore, we are in optimal position if we were to sign a qualifying FA.
Now, should we sign Choo? In the short-term, he's definitely an upgrade. With Fowler hitting lead off, you can bat Choo third and have two above-average OBP guys at the top of the lineup. You also have balance with a switch-hitter, a righty, and a lefty. Eventually, maybe Springer cleanup and Castro fifth for even more balance.
Should we look at Choo as a #3 hitter? He does have experience hitting there, though he's had mixed results:
2012-154 AB .227/.339/.338 3 HR 20 BB 42 K
2011-218 AB .248/327/.399 7 HR 23 BB 58 K
2010-395 AB .306/.412/.468 13 HR 63 BB 93 K
2009-195 AB .287/.377/.431 4 HR 19 BB 48 K
2008-70 AB .343/.420/.614 5 HR 9 BB 14 K
You would hope for 2010 production, and really would need that to justify the contract he'll receive, if you bat him third. His OBP is an asset batting second (or first), but you would have to shift Altuve somewhere between 7-9. Not that he has to bat at the top, but I would rather give him extra AB at this point than others.
He's 31, but I don't know how many years he'll get. Ellsbury received 7, will Choo get the same? That would give Fowler and Choo two of the outfield spots for '14 and '15, with Springer likely getting the other. Santana would probably be ready by '15, DDS possibly, though you could leave him at Corpus for a full season and OKC for most of '15. Long-term, I don't think he creates a problem.
The downside, of course, is that he'll get paid and get paid for a long time. How will he produce from 35-37? The top young guys won't need to be locked up for at least 5-6 years, so if the payroll continues going up, he shouldn't prevent them from signing or retaining guys in future seasons. He may prevent them from signing players this off-season, but maybe not long-term. But, it's just a matter of what you can palate in the later years and what you'll get for the price.