According to my sources, "dar la vuelta" means to go cruising in an automobile. So this target sentence means, "We'll see if I go cruising." (with the understanding that the speaker DOESN'T WANT to go cruising, even though he/she is being pressured to do so.)
I asked my Colombian friend and he said "vuelta" is like a "run-around." So a phrase like "no le doy vueltas" would mean something like "I don't give people the run-around" or "I don't beat around the bush." So I guess "vuelta" is the opposite of being direct or straight-forward.
la vuelta
Turn around.
'Vuelta' has various meanings. It can mean: return as in: Estoy de vuelta. I have returned. Hizo una vuelta por la pista. He did a lap around the track. Dio una vuelta por el parque. He took a walk through the park.
Vuelta a La Rioja was created in 1957.
They make a turn. They turn around.
La Vuelta de Rocha was created in 1937.
La Vuelta al nido was created in 1938.
Vuelta a la Independencia Nacional was created in 1979.
The cast of La vuelta - 2006 includes: Daniel Guerrero
La media vuelta is a Spanish production filmed in Spain.
La Vuelta de Martín Fierro was created in 1974.