Duggram. Failure to drain pops up on here every couple months. Here is a link to the last thread. OP thought tank was frozen but ended up being the dreaded poop pyramid.
https://www.rv.net/forum/index.cfm/fuseaction/thread/tid/29568489/srt/pa/pging/1/page/1.cfm
Some of the newer RVs use a remote cable to open the valve and the cable has been known to break. Are you sure the valve is opening today as it was yesterday? The thread above speaks to several methods to clear a blockage. I don't know if Drano will affect the components in your system and I doubt there is a pre-tank. Not knowing your trailer, the broom handle may be stopping at an elbow in the system or hitting the bottom of the tank. Either way, its a ****-shoot (just had to work that in) that any snake will find its way to the exit of the tank unless its inserted from the drain end.
WRT filling, as Maillemaker mentioned. The only ways to fill the black tank are rain coming down the vent pipe, the flush valve on the toilet not fully turning off and running into the tank, or on some RV's the bathroom sink drains to the black tank and that valve isn't fully turning off.
In the big scheme of things, getting the black tank to drain may be the bigger priority.