Everything works very well, but on some frequencies I hear loud AM broadcast stations. I think this is caused by harmonics from the VFO, but I am not sure.
What you're possibly hearing is intermodulation distortions created by your receiver from the mixer and amplifier. A local AM broadcasting station has high radiated power, making the problem even worse.
The first trouble is when you apply a pure sine wave and a pure oscillator to a mixer. In an ideal mixer, when an RF input (e.g. an RF signal from the antenna), and a LO input (e.g. a VFO) are applied to a mixer, the sum and difference frequencies appears at its output.
- $ f_\text{RF} \pm f_\text{LO} $
But in a real mixer, due to its nonlinearity, a wide range of frequency products appear at its output,
- $ m f_\text{RF} \pm n f_\text{LO} $
Two RF signals, $ f_\text{RF1} $ and $ f_\text{RF2} $ (e.g. signals from two radio stations) at different frequencies can create similar troubles. Due to the nonlinearity of amplifiers and mixers, they also create unwanted modulation between themselves.
- $ m f_\text{RF1} \pm n f_\text{RF2} $
In a receiver, the intermodulation from two RF input signals created by an amplifier can be modulated further by the local oscillator, creating even more unwanted frequencies.
- $ c(a f_\text{RF1} \pm b f_\text{RF2}) \pm d f_\text{LO}$
These higher-order intermodulation products always exist, but normally as the order goes up, their power decreases rapidly. The worst situation occurs when an amplifier or a mixer is overwhelmed by a powerful signal, these otherwise hard-to-see intermodulation products suddenly appear all over the spectrum.
The following example is a two-tone test of a RF amplifier, showing how it will create significant intermodulation when the input signal power is excessive.

Source: Wikimedia Commons, by Ice Ardor, License: CC BY-SA 4.0
In amateur radio, powerful radio stations at different bands often create a lot of troubles. The three common culprits are:
Local AM, Mediumwave radio stations
Local FM, VHF radio stations
Local/Remote AM, Shortwave Radio stations
In general, all radio receivers are vulnerable to this problem, especially when a bandpass filter or preselector is not used. A ultra-wideband direct-conversion software defined radio, like a RTL-SDR, is the most vulnerable radio receiver. Severe interference from a "blocker" radio station it can make a radio receiver become utterly useless, you'll see radio stations everywhere, and nothing else.
The first step to solve the problem is making a bandpass filter for your band-of-interest to protect your receiver from being overloaded by unwanted out-of-band signals. You can make a bandpass filter for the 80-meter band, for example.
But in this case, only the AM stations are creating the troubles, you can as well making a high-pass filter to remove all frequencies below 2 MHz, it's easier to make than a bandpass filter, as lower-Q inductors are acceptable.