I'm sorry you're dealing with this.
I don't know what is causing your dizziness and balance issues - some kind of post viral syndrome would be my guess, however the length of time it has gone on and the fact you are unable to work warrants referral for investigation of this, with either ENT or neurology.
Regarding the bleeding - you have not done anything to cause this, taking hormone therapy for a few weeks doesn't permanently alter your cycle and any effects of this will be long gone and neither here nor there at this point.
Irregular bleeding can be common in perimenopause however 2 months of continuous bleeding in a 50 year old is not normal and your GP should refer you for gynaecological investigation.
You should also have had bloods done, including a full blood count and ferritin as you may well be heading towards anaemia.
In terms of immediate bleed control, I don't love tranexamic acid, either the mechanism of action or that it doesn't address the cause of the problem, however it is worth taking to get out of a hole if that is all you have, pending investigation.
Longer term I would consider either a mirena IUS or double dose daily desogestrel ie 2*75mcg cerazette/cerelle pills for definitive bleed control.
Transdermal estrogen as a patch or gel can be used alongside either of these.