When washing machine empties, water backs up into tub. If it happens to be raining outside, the shower drains slowly and the toilet barely flushes if at all. HELP!

Are you on a septic system?

In some areas even city utilities tend to have problems handling the capacity of tremendous runoff from a heavy, or prolonged, rain. I lived in an older area of Houston, TX at one time and every time we had above normal rain the sewers would back up... toilets, sinks and shower. It's a bummer if that's your problem... but no solution other than to have the city upgrade (years) or move.

Yes we are on septic, 1000 gallon tank.

If you are on septic, you could have a filled tank and a saturated field. When was the last pump out?

Really have no idea when last pumping was. It's a rented house and I'm getting little cooperation from landlord. Full septic tank was my first thought but I really don't know much about plumbing,drains,etc. Some companies may not even come out if we don't have the septic opening uncovered(again,little cooperation from homeowner) and others will charge a lot extra, so I was hoping to have a good idea before proceeding.

Contaminated or blocked distribution lines. Rain saturated soil prevents any thing from leaving the septic tank. Need new lines.

Septic tanks are all ways full. They don't begin to work until they are full.
Pumping will not fix the problem. As soon as the tank is full again (in a few days) problem will reappear.

Need new distribution lines. That's probably why landlord doesn't want to address the problem.

Harold hit upon your problem. Your drain field can no longer disperse the liquid the septic tank sends it. Time to move the drainfield, Good luck, Tom