Is ODOT responsible for that section of Barbur Blvd? Most of that road has been relinquished to the city for maintenance.
ODOT jurisdiction starts, IIRC, at S.W. Naito Parkway and Clay Street, south along Naito to Barbur, then Barbur all the way into Tigard (Pacific Highway) and so on.
ODOT did relinquish Naito north to the Steel Bridge (however, ODOT still "maintains" the upper deck of the Steel Bridge, but not the approach spans, because the State of Oregon has a 100 year lease of that deck from Union Pacific who owns the bridge. PBOT owns the approach spans.
ODOT also relinquished Clay/Market Streets (U.S. 26) a few years ago, and I'm not sure of the current status of Harbor Drive (a.k.a. the on/off-ramp to I-5 via I-405) but the current project going on looks like it's a city project. My guess is ODOT has given it up too, at least north of Sheridan Street or possibly north of River Parkway.
As for which "route" that Barbur/Naito has...there are fewer Oregon 10 signs now than in the past. ODOT put up the 99W signs and had officially redesignated Barbur as part of 99W, only to then re-remove the designation (probably because ODOT knows it'll relinquish Barbur to PBOT at some point in the near future). But why the 10 designation remains is beyond me...considering that the only part of 10 that is ODOT maintained is from 217 east to the Washington/Multnomah County line, and I believe that its western end might have reverted back to state jurisdiction but can't be sure of it (219 to River Road) - all of the signs and the roadway specs for the last mile or so appear to be ODOT spec. And...there is one or two ODOT "islands" in the Aloha area that strangely remain in ODOT jurisdiction, yet they have no connection to any other state highway. (According to the 2010 Traffic Volumes Table, MP 5.88 through MP 7.38 is still ODOT maintained. And it shows!)