Since we don't have the luxury of JSON parsers that are available with newer versions of SQL Server, here is an ugly approach that might do the trick if the structure of your JSON string is static. For simplicity, I assumed there are no spaces other than in the specific strings to be extracted. So you might want to adjust a few numbers (11,12,13) in this code to account for that. If you go through it, you'll see that it's basically a divide and conquer approach to getting the string we want.
with your_table as
(select '{"field_text":{"current":"This is current text","previous":"This is previous text"},"CustomerIDs":{"current":"1234","previous":""}}' as json_text)
select json_text, txt1, txt2
from your_table t1
cross apply (select charindex('"current":"',json_text) as i1) t2
cross apply (select charindex('"previous":"',json_text,(i1 + 1)) as i2) t3
cross apply (select substring(json_text,(i1+11),(i2-i1-13)) as txt1) t4
cross apply (select charindex('"previous":"',json_text) as i3) t5
cross apply (select charindex('"},',json_text,(i3 + 1)) as i4) t6
cross apply (select substring(json_text,(i3+12),(i4-i3-12)) as txt2) t7;
DEMO ON SQL SERVER 2012