According to Worley's 1911 Dallas Street Directory:
Gladstone was the 4th street East of the H&TC RR, beginning at Mill Creek [not a street, but a creek] and extending South for only one block.The address numbering was different at that time, as the 8-9 houses had addresses beginning at 1500.
In 1933-34, the street name was changed from Gladstone to Diana.
Searching on a couple of online mapping services indicates no findings for a Diana St., so it must have changed again later.
The url below is for the 1920 Auto Blue Book Map of Dallas.The H&TC RR is in the center, running roughly N to S and then going toward the SE.That should give a general idea of where the street was.