I think the rule was 3 months free visa within a 6 month time period but it could be a longer time period. Whatever, it is too long to just wait it out.
When I was doing the Russian visas I would regularly go to Estonia where a new visa took about a week but the embassy was a mad house and I usually paid to have my visa done with a tourist agency.
From Ukraine you can get air tickets to Latvia for around $400 right now and hotel will cost about $50 a day. I just finished that junket because they made me leave the country to get re-registered. Such is the problem with having a double entry visa, uvir does not like to register if they do not have to and they like to tell you to just leave and come back in. One problem in this scenario is that there are times when a uvir registration is needed and just the border stamp is not good enough.
I have also done the Romania visa route by train several times. I just cannot handle the two or three day trip anymore, it really sucks. Also, the embassy in Romania has a very long line that starts early (and add the usual holiday or three) so it definately can take a solid week to 10 days to get a new visa. On the up side, there are many good hostels for $10-$15 a day and Romania is a nice place to visit.
The bottom line is that you need to leave and get a real tourist visa. Your free time is finished. They gave me just a slight hard time getting my next tourist visa. They wanted to grill me on why 3 months was not long enough. I told them I was retired and just not finished touring the country. The man said something about "women" but I had not problems.
If you decide later to get a business visa for a real business then go for a single year and not double or multiple. The year goes by quickly and uvir will stamp you after 6 months which is needed if you want the cheapest business lisence "flat tax" "sole trade". Otherwise you pay a lot of taxes with a sole enterprise.
Hope you have the time and money to do the new visa. Romania is the cheapest and you can likely do it under 10 days and $500. For about $2500 you can do the Thailand scenic route. The embassy there is very quiet and polite. Poland is a bit expensive but has some ups and downs. I have not tried Hungary. Those are just about your only close options I think.