A half-wave end-fed wire, fed against a ground plane or similar
counterpoise, will show a high impedance at (and near) resonance. If
the antenna is fed through some length of line from the "VSWR
detector," the impedance presented to the detector will be modified by
that length of line; a 1000 ohm antenna might reflect 3 ohms through
the line. But why would you expect a drop in indicated VSWR at
resonance? -- When you say "when the radiated power increases," are
you talking about increased field strength as measured by the spectrum
analyzer, with a constant output setting from the generator? How far
away is the spectrum analyzer antenna, in wavelengths, from the antenna
you are trying to measure?

If, as Cecil suspects, the antenna is really a center-fed half-wave,
what have you done to decouple the feedline from the antenna? If the
feedline comes coaxially out the end of the antenna, it may well be
that the feedline is very poorly decoupled.

Cheers,
Tom