Assalaamuaikum wa rahmatullaah
The sisters mashaAllaah have already given great advice and tips. I have also washed by hand in early marriage for a good amount of time, we couldnt afford a washing machine and neither had the space for it.
I would fill the tub with hot water and add my usual washing tablets/powder. I would give them a good mix and rub and leave to soak for awhile. I would then drain the water and scrub individual items with stains such as socks then fill the tub a little again with hot water to rinse. When it came to wringing them out - the hardest part, my husband would help me and do the larger heavy items, he was better at getting more water out as obviously he is stronger, so the clothes would dry faster.
Lingerie i would do in the sink as they are more delicate.
We would hang them on the radiators to dry.
Alhamdulillaah it is is possible, and im glad i experienced it. Just be positive! New washing machines are expensive so maybe a second hand one is better in terms of cost. Ive never had my own one, im using my mums now whilst she is away and previously I had my landlords which broke before we left (it never got the stains out anyway) so we had to use the launderette, and before that i was with my in laws!
As for your current machine, I would probably chuck it too unless it wont cost too much to repair, and search for great cleaning tips to get rid of mould. I know 1 part bleach 1 part water sprayed on it over night and then scrubbed down works well, but tiles might need re-grouting or filling around sinks etc. (I dont know what your laundry room is like!)
InshaAllaah everything turns out ok and for the best! And you can hapily do your laundry..