I have seen the relay cause intermittent operation of the OD. Check for corrosion on the fuse socket (both ends) an make sure there is no rust, green oxidation etc. on those contacts. Also, the contacts inside the relay can get tarnished as well. If you feel comfortable doing so, remove the cover and check them - clean with carbon tetrachloride or light sand paper. Either of the two issues I mentioned would result in a symptom where the relay would click but not pass current to the solenoid. Also, make sure your battery is fully charged. The solenoid takes about 17 amps to pull it in so if the battery is weak at all, the first component to fail will be the overdrive.
Regarding the fuse blowing; Two things will cause a fuse to blow- a short or an overload. You can argue that it's only one as the short causes an overload. The relay itself would not be the cause of the fuse blowing as the relay simply closes a set of contacts to allow current flow through the solenoid. If the fuse is blowing, it's likely the solenoid is causing an overload. There is a set of contacts inside the solenoid which open as it pulls in.
If these contacts are dirty and causing an arc, that can blow the fuse as it would cause an overload. Same method, clean the contacts with light sandpaper.
Another thing, if lube from the transmission gets into the solenoid, it can create suction and cause the armature to pull in slowly meaning it will take longer for those contacts to open. This too can cause the cause the fuse to blow as the contacts being closed too long creates an overload. Make sure the solenoid is clean and free of oil/grease inside where the armature moves. If it is full of grease (gear lube) you'll want to check the seal on the trans where the solenoid goes into it.