swap Shimano Alivio gear indicator for cover plate
Nope - there is no cover option. You can live without the indicators - I don't think I have ever looked at mine in earnest. Given the situation you can just tape it up as best you can to make sure none of the internals are exposed to the elements.  

There are plenty of other shifter options that all cost money that will be considerable compared to the 'OEM' cost that your cycle manufacturer paid.  

If you are willing to spend some money then you can get some very fancy Shimano shifter/brake lever combos that will have the required ergonomics. Or you may want to consider the 'unfashionable' gripshift option. Gripshift are fantastic for everyday riding and compatible with Shimano 8 speed (which I presume your Alivio setup is). People do get rid of the gripshift and put the parts on ebay. They are frequently found on 'affordable' kids bikes and can vary externally in condition, however the internals are invariably sound as there are no moving parts. Give them a go, they are fun!