This postgrad research I found offers definitions of both verbal suffixes of 渋る and 惜しむ, which seem to both mean "to hesitate to do X" when attached the masu stem of a verb:
渋る
- 動作主が~して対象を自らの外に出すことを躊躇し、出し惜しみすることを表す。
例文:(10) 銀行が企業に金を貸し渋る。- 動作主が~するのを嫌がって躊躇し、なかなか出ようとしないことを表す。
例文:(11) 子供が学校に行き渋る。惜しむ
- 動作主が自分の身から対象が出て行くのを嫌がることを表す。
例文:(10) 彼は寄付金を出し惜しむ。
The paper also compared different verbs these two suffixes attach to measuring hits from https://www.goo.ne.jp/ to see the relative frequency and kind of verb the two suffixes attached to.
The paper makes note that in the case of 惜しむ, there were way fewer hits outside of when its attached to 売る, 使う, and 出す (and 負ける for a separate meaning) compared to 渋る (even though theoretically they are similar enough to be used interchangeably).
The paper also suggests that a major difference is that 渋る focuses more on the unfavourability of giving money, using efforts, etc 「出の悪さ」and that 惜しむ focuses more on the wastefulness of using money/efforts 「もったいない」.
My issue is as follows: is there really a difference here?
I mean, the reason one would usually hesitate to lend money to someone else is well, because it would be a waste of money, which is unfavorable? This seems like a distinction without a difference.
So, outside of how it seems 渋る is more common to be attached to more verbs, looking at the tables in the paper I see no real patterns for the kinds of verbs each respective suffix tends to attach to, suggesting that this theoretical difference does not actually have a meaningful impact on how these verbs are used.
Is there any real differences between these verbal suffixes? When I want to say "I am hesitating going to school/buying that car/using our food supplies" with one of these suffixes (not ためらう or 躊躇する), is there any reason you'd use one over the other?