So if you can set the date manually, do so underground. However, it may be difficult to get it to stop fixating on the future while in range of a GPS signal.
If there is a way to manually set the date on the device, that may fix it. The problem must be that the GPS time signal is ambiguous to 1024 weeks and your Garmin thinks it’s in the wrong epoch. (That it interprets 35 as 1935 rather than 2035 when writing the GPX file just confirms that Garmin’s left hand doesn’t know what the right hand is doing.) Your device doesn’t think it’s December 1935 it thinks it’s December 2035! Which is … precisely 1024 weeks (2^10 weeks) in the future.
I have an idea about what’s going on here.