Correlation between hospital units and length of stay - r

Question

I am trying to determine if the hospital I work at should open a new unit for admissions. I intend to do this by correlating patient assigned unit and length of stay in days.
So far, I have determined that the continuous dependent variable (length of stay in days) should be log transformed, as well as the continuous age variable. Unit is categorical and contains 7 categories (unit 65, unit 66, unit 67, unit 75, unit 76, unit 77, unit 94). Gender is standard 0=male and 1=female.
My regression equation is as follows:
log_glm <- glm(log(days)~ gender + log(age) + unit, family=gaussian, data=data2022)
Model Summary:
glm(formula = log(days) ~ gender + log(age) + unit, family = gaussian,
data = data2022)
Deviance Residuals:
Min 1Q Median 3Q Max
-4.3606 -0.6712 0.0011 0.7146 2.8338
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.7416 0.9994 0.742 0.459064
gender 0.3257 0.2082 1.564 0.119566
log(age) 1.3809 0.2353 5.870 2.17e-08 ***
unitunit_66 0.1329 0.2994 0.444 0.657717
unitunit_67 1.2518 0.3334 3.755 0.000237 ***
unitunit_75 -1.8108 0.3067 -5.903 1.84e-08 ***
unitunit_76 0.2797 0.3120 0.896 0.371355
unitunit_77 -0.3540 0.3178 -1.114 0.266803
unitunit_94 0.1876 0.5350 0.351 0.726339
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for gaussian family taken to be 1.145772)
Null deviance: 429.95 on 181 degrees of freedom
Residual deviance: 198.22 on 173 degrees of freedom
(14 observations deleted due to missingness)
AIC: 552.03
Number of Fisher Scoring iterations: 2
All units at this hospital house patients with different needs. Is it appropriate to "compare" each unit in the summary output to the reference unit when all units are unique? Is this interpretation correct:
Unit 67 is statistically significant at the 0.05 level. The coefficient is 1.2518, so being assigned to unit 67 is associated with a length of stay in days 249.6631% ((exp(1.2518)-1100) higher on average than those assigned to unit 65.*
Doubting myself because that result seemed too high and interpretation incorrect, I tried a model with dummy variables for unit:
log_glm_age <- glm(log(days)~ gender + log(age) + unit_65 + unit_66 + unit_67 + unit_75 + unit_76 + unit_77 + unit_94, family=gaussian, data=data2022)
With output:
glm(formula = log(days) ~ gender + log(age) + unit_65 + unit_66 +
unit_67 + unit_75 + unit_76 + unit_77 + unit_94, family = gaussian,
data = data2022)
Deviance Residuals:
Min 1Q Median 3Q Max
-4.3606 -0.6712 0.0011 0.7146 2.8338
Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.92919 1.10243 0.843 0.400475
gender 0.32568 0.20819 1.564 0.119566
log(age) 1.38094 0.23525 5.870 2.17e-08 ***
unit_65 -0.18755 0.53500 -0.351 0.726339
unit_66 -0.05465 0.52964 -0.103 0.917936
unit_67 1.06427 0.55764 1.909 0.057979 .
unit_75 -1.99832 0.53701 -3.721 0.000268 ***
unit_76 0.09213 0.54694 0.168 0.866435
unit_77 -0.54160 0.52780 -1.026 0.306252
unit_94 NA NA NA NA
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for gaussian family taken to be 1.145772)
Null deviance: 429.95 on 181 degrees of freedom
Residual deviance: 198.22 on 173 degrees of freedom
(14 observations deleted due to missingness)
AIC: 552.03
Number of Fisher Scoring iterations: 2
But I am having doubts interpreting this. Since each patient is only admitted to one unit, is it appropriate to have all units in the model?
I don't know if it's because I am the only analyst in the entire hospital and can't discuss this with anyone who understands or if I'm having a weird day, but I can't figure out what is correct right now and could use some guidance.