How in tune is your A3? The A tone hole (opened and closed by the third finger right hand key - The "G" key) is usually the biggest factor in the pitch of A3 and F4 using the standard xox Eb | xxo fingering. If the A3 is low or stuffy at all, the pad height on that tone hole could be adjusted to open slightly more and bring the pitch of both of those notes up. This can lead to instability of other pitches if it's too far open (C#3 comes to mind) so a balance must be maintained.
Disclaimer: I am not an instrument technician, nor do I "tune" bassoons. This was a problem I had with a bassoon I once owned, although the opposite issue, the pad was opening too much and I had stability issues on C# and A3 and F4 were very sharp until I adjusted that pad.
I have never used any other fingerings for F than the "standard" fingering for it.
M.M.A., D.M.A. University of Illinois at Urbana/Champaign: B.Mus. Lawrence University
Bassoon professor at University of Wisconsin Eau-Claire
Maker of the Little-Jake electric bassoon pickup and Weasel bassoon reeds