A lot of possibilities here.
1. I have no direct experience of the Selmer but believe they are considered fair beginner instruments.
2.Bocal - as long as it has no major dents, kinks or any splits it should be OK. If it is undamaged this is last thing to change. Trying new crooks is a mind bending exercise even for experienced players & only the player can decide on one that fits the combination of him, the reeds & instrument. It is easy to make an expensive mistake - money which would have been better spent on getting the instrument & yourself in good playing condition.
3. Instrument: Lots of possibilities here. In an instrument over 30 years old you could expect deterioration of pads, regulation corks missing or worn, keys not opening correct amounts or not opening or closing when required, even if the bassoon appears to seal well. Loose fitting tenons & bocals can contribute to leaks as well. All of which can cause tuning problems. Wood rot should not be a problem if you swabbed it out properly when using it regularly.
For advice take it to a competent bassoon repairer - who will be able to give you an assessment of the condition & estimate of the cost of any work needed.
Unfortunately these are thin on the ground & many general woodwind techs are really only experienced with clarinets, flutes & saxes. The same applies to teachers - excellent players & teachers don't necessarily know anything about the mechanics of their instruments (Like many car owners who can only drive them but cannot do any simple maintenance or indeed don't notice that anything is wrong til a wheel falls off
)
4. Reed, embouchure & breathing. These are all inter-related & can contribute to tuning troubles. Really you need to find a bassoon teacher (preferably an specialist one rather than general woodwind one many of whom don't know the bassoon as a first instrument)