Try the Liberty Court hotel in Osu
The Liberty Court hotel in Osu, just down from the Mariset, is good at around 25 u.s. per night and is clean and air-conditioned. Sorry but I don't have their phone number with me but you're right about trying to book something over the phone. It can be tricky but the Liberty Court is usually not booked. I don't recommend the Mariset at around $80 u.s. per night but it is at least cheaper than La Palm. Liberty Court is near the intersection of Labadi Road and Ring Road East
Corruption is a relative concept
Cameroon is one of the most corrupt nations in the world according to Transparency International if you check out their website. Ghana, on the other hand, does not have a 'loser' President trying to rig the election rules to stay in power. So, in short, after living in Ghana for 2 years I can say that I have never had a problem in Ghana with the police as President Kufour is trying his best to 'democratize' his nation by increasing their pay and accountability. Corruption is still there but it is not as bad as Cameroon because of the leadership at the top. If you travel using local transport with the local population your chances of being harassed are also reduced.