hi Itenerant/Ardy
U have rightly pointed that the foreigner quota, it is infact available in most of the trains
but unfortunately it cant be booked online on www.irctc.co.in. Once u reach India, u can book through this quota at most of the major railway stations. Foreign tourists are relatively less in May and june so there is more probability of u getting tickets.
One more thing, if u miss this quota, then there is another quota, called "Tatkal Tickets" which can be booked in any train 5 days in advance at a slightly higher charge, this is available online.
Lot of budget airlines also operate in India on domestic network, so u can also try that option and check for the rates
Do let me know if u need more info
Arif