More importantly, Sawyer gives his time of arrival around !2:34 pm.
If Linda's identification is correct it is possible the Willis pic shows Adams out front of the TSBD before 12:34 pm
Sawyer says that he remembered a transmission on channel 2 saying that a gunman was seen in a TSBD window, but his memory is vague about the exact transmission. That would mark the latest time he was still in his car listening to the radio. The first such transmission on channel two is Haygood's at 12:35 (not 12:34):
Haygood: I just talked to a guy up here who was standing close to it and the best he could tell it came from the Texas School Book Depository Building here with that Hertz Renting sign on top.
Dispatcher: 10-4. Get his name, address, telephone number there - all the information that you can from him. 12:35 p.m.
The next is Harkness' transmission at 12:36:
Dispatcher: Attention all emergency equipment, do no use Industrial Boulevard......12:36
Harkness: I have a witness that says that it came from the 5th floor of the Texas Book Depository Store.
In the Martin film, you can see Amos Euins riding on the back of Harkness' three-wheeler eastwards from behind the pergola. They're headed towards Houston on Old Elm. In the background, in front of the TSBD entrance, a car pulls in to park. That car looks like the one that you think is Sawyer's. Harkness remembered the sequence like this:
Mr. HARKNESS - I found a little colored boy, Amos Euins, who told me he saw the shots come from that building.
Mr. BELIN - Now you just picked out a little small book, one of those little pocket notebooks?
Mr. HARKNESS - Yes, sir.
Mr. BELIN - Or a notepad from your pocket here. Is that the original notation that you made?
Mr. HARKNESS - Yes, sir.
Mr. BELIN - When did you make that notation?
Mr. HARKNESS - Immediately after the shooting.
Mr. BELIN - Is that your own record that you have kept in your possession since then?
Mr. HARKNESS - Yes, sir. I turned.
Mr. BELIN - You turned what?
Mr. HARKNESS - After I took his name and address and put this information on the radio, I then took him on the back of my three-wheel motorcycle and put him in Inspector Sawyer's car.
So if Harkness made his 12:36 transmission before he put Euins on the motorcyle, and the car in the Martin film belongs to Sawyer, then Sawyer could not have parked in front of the Depository before 12:36
BTW, this is the best online version of the Martin film I can find. Euins and Harkness (+subsequent paleophotobomb) appear about 0:50 in: