At this time there are only eight properties under $200K in the 4 cities you mentioned. Homes are selling very quickly since there are plenty of all cash buyers in this price range and the demand from investors is huge.
Are you working with an agent? Your agent can send you listings directly from the MLS of all homes that meet your criteria.
Additionally, I suspect that properties priced between $190,000 and $200,000 will be likely to be bid up to more than $200K in the very fast moving market.
Note that in addition to the purchase price, you should also factor closing costs which can vary depending on the city's transfer taxes, HOA transfer fees, etc if applicable
Do you have a realtor helping you? If not, that's the first thing you should do -- engage someone to help you find the properties to meet your needs.
I have another client in the same position --- unfortunately, not all the properties that are in his budget range meet his approval. You should consider making some allowances regarding the size, location and condition of the properties. Setting very high expectations may not help in a very competitive market that includes many cash buyers.
Let me know how I can help