I live in an older house and recently we've had a lot of rain coupled with melting snow.

At first we thought water was backing up into our plumbing, but we honestly aren't sure. Right now our downstairs toilet is backed up with water and bits of a mud-like substance (not sewage for certain) and can't flush. Our turned off dishwasher fills up with water every few hours and has to be drained by hand. The kitchen sinks will drain, but only do so slowly (takes about an hour for a full sink to empty).

Our upstairs plumbing is working fine. We can't afford to have a plumber diagnose the problem, so we need to know what it is before we call.

It is likely the whole house is clogged, that's why your dishwasher is filling up. Water/sewage from upstairs can't get out so it backs into the washer
Do you have a basement and/or a clean out cap?

Our upstairs plumbing is working fine. and the reason it works so find is that when you flush it all comes out in the first floor. You have a total blockage downstream from the first floor fixtures.
If you know where your house clean out is located then open it up and let me know if you have standing water in it. Back to you, Tom

Isn't it nice to get answers back from the OP once you have taken time to reply with qualified answers?? And they can't afford a plumbetr??

We have a basement (it's pretty wet right now). Do you think the problem's with the clean out cap?

It is likely the whole house is clogged, that's why your dishwasher is filling up. Water/sewage from upstairs can't get out so it backs into the washer
Do you have a basement and/or a clean out cap?

We have a basement (it's pretty wet right now). Do you think the problem's with the clean out cap?

and the reason it works so find is that when you flush it all comes out in the first floor. You have a total blockage downstream from the first floor fixtures.
If you know where your house clean out is located then open it up and let me know if you have standing water in it. Back to you, Tom

Thank you for the answer, I'm pretty sure that's the problem. Now that I know where the issue is I can get someone in to fix it.

If you know where your cleanout is located and wish to do the job yourself I can walk you through it . What's your pleasure? Back to you, Tom